Method for fault location in electric power lines

ABSTRACT

A method for locating faults in three-terminal and multi-terminal power lines, applicable in electric power systems for overhead and underground cable transmission and distribution lines. The inventive method is characterised in that it measures current for fault and pre-fault conditions and in one terminal station of the system line phase voltage for fault and pre-fault conditions is measured, a hypothetical fault location is assumed, the distances to the hypothetical fault locations are calculated and fault resistance is calculated, the actual fault point is selected by first comparing the numerical values concerning the distances to the hypothetical fault locations and rejecting those results whose numerical values are negative or bigger than one in relative units, and then by analysing the values of the calculated fault resistances for the fault locations and rejecting those results of the calculations for which the value of fault resistance is negative, and if there is still more than one calculation result remains not rejected then performing selection of the valid result by analysing the values of the calculated respective equivalent source impedances.

The present invention is concerned with a method for fault location inelectric power lines applicable both to three-terminal power lines andmultiple-terminal power lines, fit to be used in the power industry foroverhead and overhead-cable transmission or distribution lines.

The accurate location of the fault in electric power lines is of greatimportance both for power companies dealing with electric energydistribution and for end users of electric energy. Quick and exactlocation of the fault point effects the quality of transmitted electricenergy and its reliable and uninterrupted supply. In most cases, faultsresult in mechanical damage which must be removed before voltage isrestored in the line. Quick rectification of a fault is possible if theplace of the fault is known exactly. The simplest method for determiningthe location of a fault is searching along the line. This method is slowand expensive, even dangerous in adverse weather conditions. Faultlocators facilitating quick determination of the fault point are usedfor locating fault points. A fault locator is usually a part of adigital protective relay located in power stations or substations.Depending on the type of electric power lines: parallel lines,three-terminal power lines, multiple-terminal power lines, and dependingon the location of final terminals and the diversity of measuringsignals, different ways of fault location are distinguished.

A system and a method for fault location in a three-terminal power lineis known from U.S. Pat. No. 6,466,030. The method according to thatinvention consists in dividing the transmission line in the tap pointinto two sections, the supply side section and the receiving sidesection, and on both sides of both sections at their ends instrumentsfor measuring the values of current and voltage signals are installed.Then, on the basis of synchronously or asynchronously measured valuesand the model of fault loops, the load impedance in the branch iscalculated, after which the first hypothetical fault location iscalculated assuming that the fault occurred on the supply section side.Depending on whether the measurements are synchronised or not, eitherthe phase angle is calculated, which is the measure of displacement withtime of the measured samples from signals from both ends of the line onthe basis of measured pre-fault signals, or a phase angle equal to zerofor synchronous measurements is assumed. Then calculations of the secondhypothetical fault location in the second line section between the tappoint and the received point are made. From the two calculatedhypothetical locations one value which is contained in a specificinterval of expected values, i.e. numerical values from 0 to 1 inrelative units is chosen. The described solution applies to the case ofa single circuit line with a passive branch, which means that in theadopted equivalent circuit diagram of such system, in the tapped line,the presence of electric-power motive force is not considered, and theload impedance of this line can be calculated from pre-faultmeasurements.

A system and a method for fault location in a multiple-terminal paralleltransmission line is known from U.S. Pat. No. 5,485,394. In the methodaccording to that invention, a multi-terminal transmission system isequalled to a three-terminal transmission system. For such systemdifferential current amplitudes are calculated in each station, and thenthe distance to the fault point is calculated from their relations.

A method for fault location using voltage and current phasor measurementin all stations at the ends of a multi-terminal line is known from thepublication “Novel Fault Location Algorithm for Multi-Terminal LinesUsing Phasor Measurement Units” published in the materials of theThirty-Seventh Annual North American Power Symposium in Ames, Iowa, USA,Oct. 23-25, 2005. That method consists in reducing a multi-sectiontransmission line to systems of two-terminal lines assuming that thefault is located in one of these sections and then hypothetical faultlocations are calculated for this assumption. Next, calculations ofsuccessive hypothetical fault locations are made assuming that the faultis located in further successive sections of the line. One value, whichis contained in a specific interval of expected values and whichindicates the actual place of the fault, is selected from thehypothetical locations calculated in this way.

THE ESSENCE OF THE INVENTION

The essence of the inventive method for locating faults in electricpower lines by dividing the lines of the transmission or distributionsystem into sections and assuming the hypothetical location of the faulton at least one of these sections consists in the following:

current for fault condition and pre-fault condition is measured in allterminal stations of the system,

the line phase voltage for fault and pre-fault conditions is measured inone terminal station of the system,

the symmetrical components of the measured current and voltage signalsas well as the total fault current in the fault point are calculated,

the first hypothetical fault point located in the line section betweenthe beginning of the line and the first tap point, the secondhypothetical fault point located in the line section between the end ofthe line and the last tap point of the branch, and a successivehypothetical fault point which is located in the branch are assumed,additionally assuming successive hypothetical fault points located inthe line sections between two consecutive tap points for amulti-terminal line,

the distance from the beginning of the line to the fault point, thedistance from the end of the line to the fault point and the distancefrom the end of the tapped line to the fault point located in thisbranch are calculated and, for a multi-terminal line, the distance fromthe tap point to the fault point located in the line section between twotap points is additionally calculated, and then for all hypotheticalfault points in each section fault resistance is calculated,

the actual fault point is selected by first comparing the numericalvalues concerning the previously determined distances and rejectingthose results whose numerical values are negative or bigger than 1 inrelative units and then, by analysing the fault resistance valuescalculated for fault points and rejecting those results for which thefault resistance is negative, after which if it is found that only onenumerical value concerning the distance is contained within the intervalfrom zero to one in relative units and the value of the calculated faultresistance for this distance to the fault point is positive or equal tozero, these results are final results and they indicate the actualdistance to the fault point and the fault resistance value in the faultpoint,

if, after the selection of the actual fault point, it turns out that atleast two numerical values concerning the previously calculateddistances are contained within the numerical interval from zero to onein relative units and the values of the calculated fault resistance forthese fault points are positive or equal to zero, then impedance modulesor impedances of equivalent source systems are determined for thenegative sequence component for phase-to-ground faults, phase-to-phasefaults and phase-to-phase-to-ground faults or for the incrementalpositive sequence component for three-phase faults and on the assumptionthat the fault occurred in a definite section, and during the impedancedetermination it is additionally checked whether the calculated valuesof the impedance of equivalent source systems are contained in the firstquadrant of the Cartesian co-ordinate system for the complex plain andthese distances to fault points are rejected for which impedance valuesare not contained in this quadrant of the system, and if it turns outthat only one value of the impedance of the equivalent source systemconcerning distance is contained in the first quadrant of the system,then the result of the calculation of the distance to the fault point,for this impedance, is considered to be final, whereas if it turns outthat at least two values of the impedance of equivalent source systemsconcerning distance are contained in the first quadrant of the system,then the modules of these impedances are determined,

the values of the modules of the equivalent source impedance arecompared with realistic values, which really define the system load, andthe distance for which the value of the module of the equivalent sourceimpedance is nearest to the realistic values, really determining thesystem load, is considered to be the final result.

Preferably, calculation of the total fault current is done taking intoaccount the share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated, a specially determined set ofthese coefficients being used for that operation.

Preferably, for phase-to-phase-to-earth faults the positive sequencecomponent is eliminated in the estimation of the total fault current,and for the negative and zero sequence components the following valuesof the share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated are assumed, in particular fora-b-g fault:

$\; {{{\underset{\_}{a}}_{F\; 1} = 0},{{\underset{\_}{a}}_{F\; 2} = {{\underset{\_}{a}}_{F\; 2}^{{init}.} - \; \frac{{\underset{\_}{a}}_{F\; 1}^{{init}.}{\underset{\_}{b}}_{F\; 2}}{{\underset{\_}{b}}_{F\; 1}}}},{{\underset{\_}{a}}_{F\; 0} = \frac{{\underset{\_}{a}}_{F\; 1}^{{init}.}}{{\underset{\_}{b}}_{F\; 1}}}}$whereas:${{\underset{\_}{a}}_{F\; 1}^{{init}.} = {1 - {\underset{\_}{a}}^{2}}},{{\underset{\_}{a}}_{F\; 2}^{{init}.} = {1 - \underset{\_}{a}}},{{\underset{\_}{b}}_{F\; 1} = {- \underset{\_}{a}}},{{\underset{\_}{b}}_{F\; 2} = {- {\underset{\_}{a}}^{2}}},{\underset{\_}{a} = {\exp^{j\; 2\; {\pi \;/3}} = {- 0}}},{5 + {j\frac{\sqrt{3}}{2}}},$

Preferably, for three-terminal power lines, the distances from thebeginning of the line to the fault point d_(A), from the end of the lineto the fault point d_(B), from the end of the tapped line to the faultpoint d_(C) are determined from the following equations:

$\; {{d_{A} = \frac{{{{real}\left( {\underset{\_}{V}}_{AP} \right)}\; {{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {\underset{\_}{V}}_{AP} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}{{{{real}\left( {{\underset{\_}{Z}}_{ILA}{\underset{\_}{I}}_{AP}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{ILA}{\underset{\_}{I}}_{AP}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},{d_{B} = \frac{\begin{matrix}{{{- {real}}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{ILB}{\underset{\_}{I}}_{TBp}}} \right){{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{imag}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{ILB}{\underset{\_}{I}}_{TBp}}} \right){{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{{{{real}\left( {{\underset{\_}{Z}}_{ILB}{\underset{\_}{I}}_{TBp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{ILB}{\underset{\_}{I}}_{TBp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},{d_{C} = \frac{\begin{matrix}{{{- {real}}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{ILC}{\underset{\_}{I}}_{TCp}}} \right){{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{imag}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{ILC}{\underset{\_}{I}}_{TCp}}} \right){{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{{{{real}\left( {{\underset{\_}{Z}}_{ILC}{\underset{\_}{I}}_{TCp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{ILC}{\underset{\_}{I}}_{TCp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},}\;$

where:“real” denotes the real part of the given quantity,“imag” denotes the imaginary part of the given quantity,V_(Ap)—denotes the fault loop voltage determined assuming that the faultoccurred in section LA,V_(Tp)—denotes the fault loop voltage determined assuming that the faultoccurred in section LB or LC,I_(Ap)—denotes the fault loop current determined assuming that the faultoccurred in section LA,I_(TBp)—denotes the fault loop current determined assuming that thefault occurred in section LB,I_(TCp)—denotes the fault loop current determined assuming that thefault occurred in line section LC,I_(F)—denotes total fault current,Z_(1LA)=R_(1LA)+jω₁L_(1LA)—denotes impedance of the line section LA forthe positive sequence,Z_(1LB)=R_(1LB)+jω₁L_(1LB)—denotes impedance of the line section LB forthe positive sequence,Z_(1LC)=R_(1LC)+jω₁L_(1LC)—denotes impedance of the line section LC forthe positive sequence,R_(1LA), R_(1LB), R_(1LC)—resistance for the positive sequence for linesections LA, LB, LC, respectively,L_(1LA), L_(1LB), L_(1LC)—inductance for the positive sequence for linesections LA, LB, LC, respectively,ω₁—pulsation for the fundamental frequency.

Preferably, for three-terminal power lines, the fault resistance R_(FA),R_(FB), R_(FC) is determined from the following equations:

${R_{FA} = {\frac{1}{2}\begin{bmatrix}{\frac{{{real}\left( {\underset{\_}{V}}_{Ap} \right)} - {d_{A}{{real}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}}}{{real}\left( {\underset{\_}{I}}_{F} \right)} +} \\\frac{{{imag}\left( {\underset{\_}{V}}_{Ap} \right)} - {d_{A}{{imag}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}\end{bmatrix}}},{R_{FB} = {{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{B}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} + {\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{B}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}}$$R_{FC} = {{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{C}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} + {\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{C}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}$

where:“real” denotes the real part of the given quantity,“imag” denotes the imaginary part of the given quantity,V_(Ap)—denotes the fault loop voltage calculated assuming that the faultoccurred in section LA,V_(Tp)—denotes the fault loop voltage calculated assuming that the faultoccurred in section LB or LC,I_(Ap)—denotes the fault loop current calculated assuming that the faultoccurred in section LA,I_(TBp)—denotes the fault loop current calculated assuming that thefault occurred in section LB,I_(TCp)—denotes the fault loop current calculated assuming that thefault occurred in line section LC,I_(F)—denotes the total fault current,Z_(1LA)=R_(1LA)+jω₁L_(1LA)—denotes impedance of the line section LA forthe positive sequence,Z_(1LB)=R_(1LB)+jω₁L_(1LB)—denotes impedance of the line section LB forthe positive sequence,Z_(1LC)=R_(1LC)+jω₁L_(1LC)—denotes impedance of the line section LC forthe positive sequence,R_(1LA), R_(1LB), R_(1LC)—resistance for the positive sequence for linesections LA, LB, LC, respectively,L_(1LA), L_(1LB), L_(1LC)—inductance for the positive sequence for linesections LA, LB, LC, respectively,ω₁—pulsation for the fundamental frequency.d_(A)—denotes the distance from the beginning of the line to the faultpoint,d_(B)—denotes the distance from the end of the line to the fault point,d_(C)—denotes the distance from the end of the tapped line to the faultpoint.

Preferably, for three-terminal power lines, the equivalent sourceimpedance for the negative sequence component ((Z_(2SB))_(SUB) _(—)_(A)) and for the incremental positive sequence component((Z_(Δ1SB))_(SUB) _(—) _(A)) are calculated assuming that the faultoccurred in LA line section, as per this equation:

$\left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ A} = {\frac{{{\underset{\_}{G}}_{iA}{\underset{\_}{I}}_{A\; 2}} - {{\underset{\_}{H}}_{iA}{\underset{\_}{I}}_{FAi}}}{{\underset{\_}{Q}}_{BCi}\left( {{\underset{\_}{I}}_{FAi} - {\underset{\_}{I}}_{Ai}} \right)}.}$

where:the lower index i takes on values i=2 for the negative sequence, i=Δ1for the incremental positive sequence component,G_(iA)—denotes the first analytical coefficient for the negativesequence component, determined from the analysis of an equivalentcircuit diagram of the system as shown in FIG. 11 and/or for theincremental positive sequence component analytically determined from theequivalent circuit diagram of the system as shown in FIG. 12,I_(Ai)—denotes the negative and/or incremental positive sequencecomponent of current measured at the beginning of the line,H_(iA)—denotes the second analytical coefficient for the negativesequence component, determined from the analysis of an equivalentcircuit diagram of the system as shown in FIG. 11 and/or the incrementalpositive sequence component analytically determined from the equivalentcircuit diagram of the system as shown in FIG. 12,I_(FAi)—denotes the negative sequence component of the total faultcurrent, determined from the analysis of an equivalent circuit diagramof the system as shown in FIG. 11 and/or the incremental positivesequence component of the total fault current, determined from theanalysis of an equivalent circuit diagram of the system as shown in FIG.12,Q_(BCi)—denotes the quotient of the negative sequence component ofcurrent measured at the end of the line and the sum of the negativesequence components of current signals measured at the end of the lineand at the end of the tapped line and/or the quotient of the incrementalpositive sequence component of current measured at the end of the lineand the sum of the incremental positive sequence components of currentsignals measured at the end of the line and at the end of the tappedline.

Preferably, for three-terminal power lines, the equivalent sourceimpedance ((Z_(2SC))_(SUB) _(—) _(A)) for the negative sequencecomponent and ((Z_(Δ1SC))_(SUB) _(—) _(A)) for the incremental positivesequence component are calculated assuming that the fault occurred inline section LA, from the following equation:

$\left( {\underset{\_}{Z}}_{iSC} \right)_{SUB\_ A} = {{\left( {{\underset{\_}{Z}}_{iLB} + \left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ A}} \right)\frac{{\underset{\_}{I}}_{Bi}}{{\underset{\_}{I}}_{Ci}}} - {{\underset{\_}{Z}}_{iLC}.}}$

where:the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,(Z_(iSB))_(SUB) _(—) _(A)—denotes equivalent source impedance for thenegative sequence component and/or the incremental positive sequencecomponent, calculated assuming that the fault occurred in line sectionLA,Z_(iLB)—denotes the impedance of line section LB for the negativesequence component and/or the positive sequence component, where:Z_(Δ1LB)=Z_(1LB),Z_(1LB)—denotes the impedance of line section LB for the positivesequence component,Z_(iLC)—denotes the impedance of line section LC for the negativesequence component and/or impedance of line section LC for theincremental positive sequence component, where Z^(2LC)=Z_(1LC) andZ_(Δ1LC)=Z_(1LC),Z_(1LC)—denotes the impedance of line section LC for the positivesequence component,I_(Bi)—denotes the negative sequence component and/or the incrementalpositive sequence component of current measured at the end of the line,I_(Ci)—denotes the negative sequence component and/or the incrementalpositive sequence component of current measured at the end of thebranch.

Preferably, for three-terminal power lines, equivalent source impedancefor the negative sequence component (Z_(2SB))_(SUB) _(—) _(B) and forthe incremental positive sequence component (Z_(Δ1SB))_(SUB) _(—) _(B)is determined assuming that the fault occurred in line section LB, fromthe following equation:

$\left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ B} = \frac{{\left( {1 - d_{B}} \right){\underset{\_}{Z}}_{iLB}{\underset{\_}{I}}_{TBi}^{{transf}.}} - {d_{B}{\underset{\_}{Z}}_{iLB}{\underset{\_}{I}}_{Bi}} - {\underset{\_}{V}}_{Ti}^{{transf}.}}{{\underset{\_}{I}}_{Bi}}$

where:the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,d_(B)—denotes the distance from the end of the line to the fault point,Z_(iLB)—denotes the impedance of line section LB for the negativesequence component and/or for the positive sequence component, whereZ_(2LB)=Z_(1LB) and Z_(Δ1LB)=Z_(1LB),Z_(1LB)—denotes the impedance of line section LB for the positivesequence component,I_(TBi) ^(transf.)—denotes current flowing from a tap point T to linesection LB for the negative sequence component and/or for theincremental positive sequence component,I_(Bi)—denotes the negative sequence component and/or the incrementalpositive sequence component of current measured at the end of the line,V_(Ti) ^(transf.)—denotes voltage in the tap point T for the negativesequence component and/or for the incremental positive sequencecomponent.

Preferably, for three-terminal power lines, equivalent source impedancefor the negative sequence component (Z_(2SC))_(SUB) _(—) _(B) and forthe incremental positive sequence component (Z_(Δ1SC))_(SUB) _(—) _(B)is calculated assuming that the fault occurred in line section LB, fromthe following equation:

$\left( {\underset{\_}{Z}}_{iSC} \right)_{SUB\_ B} = {- \frac{{\underset{\_}{V}}_{Ci}}{{\underset{\_}{I}}_{Ci}}}$

where:the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,V_(Ci)—denotes the calculated negative sequence component and/or theincremental positive sequence component of voltage at the end of thetapped line,I_(Ci)—denotes the negative sequence component and/or the incrementalpositive sequence component of current measured at the end of thebranch.

Preferably, for three-terminal power lines, equivalent source impedancefor the negative sequence component (Z_(2SC))_(SUB) _(—) _(C) and forthe incremental positive sequence component (Z_(Δ1SC))_(SUB) _(—) _(C)is calculated assuming that the fault occurred in line section LC, fromthe following equation:

$\left( {\underset{\_}{Z}}_{iSC} \right)_{SUB\_ C} = \frac{{\left( {1 - d_{C}} \right){\underset{\_}{Z}}_{iLC}{\underset{\_}{I}}_{TCi}^{{transf}.}} - {d_{C}{\underset{\_}{Z}}_{iLC}{\underset{\_}{I}}_{Ci}} - {\underset{\_}{V}}_{Ti}^{{transf}.}}{{\underset{\_}{I}}_{Ci}}$

where:the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,d_(C)—denotes the distance from the end of the tapped line to the faultpoint,Z_(iLC)—denotes impedance of the line section LC for the negativesequence component and/or for the incremental positive sequencecomponent, where Z_(2LC)=Z_(1LC) and Z_(Δ1LC)=Z_(1LC),Z_(1LC)—denotes impedance of the line section LC for the positivesequence component,I_(TCi) ^(transf.)—denotes current flowing from the tap point T to linesection LC for the negative sequence component and/or for theincremental positive sequence component,I_(Ci)—denotes the negative sequence component and/or the incrementalpositive sequence component of current measured at the end of thebranch,V_(Ti) ^(transf.)—denotes voltage at the tap point T for the negativesequence component and/or for the incremental positive sequencecomponent.

Preferably, for three-terminal power lines, equivalent source impedancefor the negative sequence component (Z_(2SB))_(SUB) _(—) _(C) and forthe incremental positive sequence component (Z_(Δ1SB))_(SUB) _(—) _(C)is calculated from the following equation assuming that the faultoccurred in line section LC:

$\left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ C} = {- \frac{{\underset{\_}{V}}_{Bi}}{{\underset{\_}{I}}_{Bi}}}$

where:the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,V_(Bi)—denotes the calculated negative sequence and/or the incrementalpositive sequence component of voltage at the end of the line,I_(Bi)—denotes the negative sequence and/or the incremental positivesequence component of current measured at the end B of the line.

Preferably, for multi-terminal power lines, distances from the beginningof the line to the fault point (d₁), from the end of the line to thefault point (d_((2n−3))), from the end of the line to the fault point(d_((2k−2))), from the tap point to the fault point in the line sectionbetween two tap points (d_((2k−1))) are determined from the followingequations:

${d_{1} = \frac{{{{real}\left( {\underset{\_}{V}}_{1p} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {\underset{\_}{V}}_{1p} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}{{{{real}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},{d_{({{2n} - 3})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}}$ ${d_{({{2k} - 2})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tkkp} - {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{Tkkp} - {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}},{d_{({{2k} - 1})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}}} \right)}{real}*\left( {\underset{\_}{I}}_{F} \right)}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}}$

where:“real” denotes the real part of the given quantity,“imag” denotes the imaginary part of the given quantity,V_(1p)—fault loop voltage calculated under assumption that faultoccurred in the first section of the line section L1,I_(1p)—fault loop current calculated under assumption that faultoccurred in the first section of the line section L1,V_(T(n−1)np)—fault loop voltage calculated under assumption that faultoccurred in the line section L(2n−3),I_(T(n−1)np)—fault loop current calculated under assumption that faultoccurred in the line section L(2n−3),V_(Tkkp)—fault loop voltage calculated under assumption that faultoccurred in the k^(th) tapped line,I_(Tkkp)—fault loop current calculated under assumption that faultoccurred in k^(th) tapped line,V_(TkT(k+1)p)—fault loop voltage calculated under assumption that faultoccurred in the line section between two tap points,I_(TkT(k+1)p)—fault loop current calculated under assumption that faultoccurred in the line section between two tap points,I_(F)—total fault current,Z_(1L1)—impedance of line section L1 for the positive sequencecomponent,Z_(0L1)—impedance of line section L1 for the zero sequence component,Z_(1L(2n−3))—impedance of line section L(2n−3) for the positive sequencecomponent,Z_(0L(2n−3))—impedance of line section L(2n−3) for the zero sequencecomponent,Z_(1L(2k−2))—impedance of line section L(2k−2) for the positive sequencecomponent,Z_(0L(2k−2))—impedance of line section L(2k−2) for the zero sequencecomponent,Z_(1L(2k−1))—impedance of line section L(2k−1) for the positive sequencecomponent,Z_(0L(2k−1))—impedance of line section L(2k−1) for the zero sequencecomponent.k—number of the tap pointn—number of the line terminal

Preferably, for multi-terminal power lines, the fault resistance(R_(1F)), (R_((2n−3)F)), (R_((2k−2)F)), (R_((2k−1)F)) is calculated fromthe following equations:

$\mspace{79mu} {{R_{1F} = {\frac{1}{2}\begin{bmatrix}{\frac{{{real}\left( {\underset{\_}{V}}_{1p} \right)} - {d_{1}{{real}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}}}{{real}\left( {\underset{\_}{I}}_{F} \right)} +} \\\frac{{{imag}\left( {\underset{\_}{V}}_{1p} \right)} - {d_{1}{{imag}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}\end{bmatrix}}},{R_{{({{2m} - 3})}F} = {{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{{T{({n - 1})}}{np}} \right)} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}++}{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{{T{({n - 1})}}{np}} \right)} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}},{R_{{({{2k} - 2})}F} = {{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tkkp} \right)} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}++}{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tkkp} \right)} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}}}$$R_{{({{2k} - 1})}F} = {{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} \right)} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}++}{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} \right)} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}$

where:“real” denotes the real part of the given quantity,“imag” denotes the imaginary part of the given quantity,d₁—distance to fault from the beginning of the line to the fault point,d_((2n−3))—distance to fault from the end of the line to the faultpoint,d_((2k−2))—distance to fault from the end of the tapped line to thefault pointd_((2k−1))—distance to fault on the line section between two tap points,V_(1p)—fault loop voltage calculated under assumption that faultoccurred in the first section of the line section L1,I_(1p)—fault loop current calculated under assumption that faultoccurred in the first section of the line section L1,V_(T(n−1)np)—fault loop voltage calculated under assumption that faultoccurred in the line section L(2n−3),I_(T(n−1)np)—fault loop current calculated under assumption that faultoccurred in the line section L(2n−3),V_(Tkkp)—fault loop voltage calculated under assumption that faultoccurred in the k^(th) tapped line,I_(Tkkp)—fault loop current calculated under assumption that faultoccurred in k^(th) tapped line,V_(TkT(k+1)p)—fault loop voltage calculated under assumption that faultoccurred in the line section between two tap points,I_(TkT(k+1)p)—fault loop current calculated under assumption that faultoccurred in the line section between two tap points,I_(F)—total fault current,Z_(1L1)—impedance of line section L1 for the positive sequencecomponent,Z_(0L1)—impedance of line section L1 for the zero sequence component,Z_(1L(2n−3))—impedance of line section L(2n−3) for the positive sequencecomponent,Z_(0L(2n−3))—impedance of line section L(2n−3) for the zero sequencecomponent,Z_(1L(2k−2))—impedance of line section L(2k−2) for the positive sequencecomponent,Z_(0L(2k−2))—impedance of line section L(2k−2) for the zero sequencecomponent,Z_(1L(2k−1))—impedance of line section L(2k−1) for the positive sequencecomponent,Z_(0L(2k−1))—impedance of line section L(2k−1) for the zero sequencecomponent.k—number of the tap pointn—number of the line terminal

Preferably, for multi-terminal power lines, equivalent source impedancefor the negative sequence component (Z_(2S1)) or for the incrementalpositive sequence component (Z_(Δ1S1)) is calculated assuming that thefault is located in the line section between the beginning of the lineand the first tap point, according to the following equation:

$\left( {\underset{\_}{Z}}_{{iS}\; 1} \right) = \frac{- {\underset{\_}{V}}_{1i}}{{\underset{\_}{I}}_{1\; i}}$

wherei=2 for negative sequence, i=Δ1 for incremental positive sequencecomponent,V_(1i)—voltage measured in station 1 (the first lower index) forindividual symmetrical components, (the second lower index) i.e.negative component—index 2 and incremental positive sequencecomponent—index Δ1,I_(1i)—current measured in station 1 (the first lower index) forindividual symmetrical components, (the second lower index) i.e.negative component—index 2 and incremental positive sequencecomponent—index Δ1.

Preferably, for multi-terminal power lines, equivalent source impedance((Z_(2S(n)))) for the negative sequence component and ((Z_(Δ1S(n)))) forthe incremental positive sequence component is determined assuming thatthe fault is located in the line section between the end of the line andthe last tap point, from the following equation:

$\left( {\underset{\_}{Z}}_{iSn} \right) = {- {\frac{\begin{matrix}{{\underset{\_}{V}}_{{T{({n - 1})}}i}^{{transf}.} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2n} - 3})}} \cdot {\underset{\_}{I}}_{{T{({n - 1})}}{ni}}^{{transf}.}} -} \\{d_{({{2n} - 3})} \cdot {\underset{\_}{Z}}_{{iL}{({{2n} - 3})}} \cdot \left( {{\underset{\_}{I}}_{{T{({n - 1})}}{ni}}^{{transf}.} - {\underset{\_}{I}}_{Fi}} \right)}\end{matrix}}{{\underset{\_}{I}}_{ni}}.}}$

where:i=2 for negative sequence component, i=Δ1 for incremental positivesequence component,V_(T(n−1)i) ^(transf.)—voltages in the final tap point T(n−1) for thenegative sequence component i=2, or for incremental positive sequencecomponent i=Δ1,d_((2n−3))—distance to fault from the end of the line to the faultpoint,Z_(iL(2n−3))—impedance of line section L(2n−3) for the negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1,I_(T(n−1)ni) ^(transf.)—values of current flowing from tap point T(n−1)to station n in line section L(2n−3) for negative sequence componenti=2, or for incremental positive sequence component i=Δ1I_(Fi)—total fault current for negative sequence component i=2, or forincremental positive sequence component i=Δ1,I_(ni)—current measured in last station n (the first lower index) forindividual symmetrical components, (the second lower index) i.e.negative component—index 2 and incremental positive sequencecomponent—index Δ1.

Preferably, for multi-terminal power lines, equivalent source impedancefor the negative sequence component ((Z_(2Sk))) and for the incrementalpositive sequence component ((Z_(Δ1Sk))) is determined assuming that thefault is located in the tapped line, from the following equation:

$\left( {\underset{\_}{Z}}_{iSk} \right) = {- \frac{\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{transf}.} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 2})}} \cdot {\underset{\_}{I}}_{Tkki}^{{transf}.}} -} \\{d_{({{2k} - 2})} \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 2})}} \cdot \left( {{\underset{\_}{I}}_{Tkki}^{{transf}.} - {\underset{\_}{I}}_{F\; 2}} \right)}\end{matrix}}{{\underset{\_}{I}}_{ki}}}$

where:V_(Tki) ^(transf.)—voltages in the k^(th) tap point for, the negativesequence component i=2, or for incremental positive sequence componenti=Δ1,d_((2k−2))—distance to fault from the end of the tapped line to thefault point Tk,Z_(1L(2k−2))—impedance of line section L(2k−2) for the negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1,I_(Tkki) ^(transf.)—values of current flowing from tap point Tk tok^(th) station in tapped line section L(2k−2) for negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1I_(Fi)—total fault current for negative sequence component i=2, or forincremental positive sequence component i=Δ1,I_(ki)—current measured in station k (the first lower index) forindividual symmetrical components, (the second lower index) i.e.negative component—index 2 and incremental positive sequencecomponent—index Δ1.

Preferably, for multi-terminal power lines, equivalent source impedancefor the negative sequence component ((Z_(2Sk)) and (Z_(2S(k+1)))) andfor the incremental positive sequence component ((Z_(Δ1Sk)) and(Z_(Δ1S(k+1)))) is calculated assuming that the fault is located in theline section between two consecutive tap points, from the followingequations:

${\left( {\underset{\_}{Z}}_{iSk} \right) = {- \frac{\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{transf}.} - {d_{({{2k} - 1})} \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot {\underset{\_}{I}}_{{{TkT}{({k + 1})}}i}^{{transf}.}} -} \\{{\left( {1 - d_{({{2k} - 1})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot \left( {{\underset{\_}{I}}_{{{TkT}{({k + 1})}}i}^{{transf}.} - {\underset{\_}{I}}_{Fi}} \right)} + {{\underset{\_}{Z}}_{iLk}{\underset{\_}{I}}_{ki}}}\end{matrix}}{{\underset{\_}{I}}_{ki}}}},{\left( {\underset{\_}{Z}}_{{iS}{({k + 1})}} \right) = {- \frac{\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{transf}.} - {d_{({{2k} - 1})} \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot {\underset{\_}{I}}_{{{TkT}{({k - 1})}}i}^{{transf}.}} -} \\{{\left( {1 - d_{({{2k} - 1})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot \left( {I_{{{TkT}{({k + 1})}}i}^{{transf}.} - {\underset{\_}{I}}_{Fi}} \right)} + {{\underset{\_}{Z}}_{{iL}{({2k})}}{\underset{\_}{I}}_{{({k + 1})}i}}}\end{matrix}}{{\underset{\_}{I}}_{{({k + 1})}i}}}}$

where:V_(Tki) ^(transf.)—voltages in the k^(th) tap point for the negativesequence component i=2, or for incremental positive sequence componenti=Δ1,d_((2k−1))—distance to fault on the line section between two tap pointsZ_(1L(2k−2))—impedance of line section L(2k−1) for the negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1,I_(TkT(k+1)i) ^(transf.)—current flowing from tap point Tk to tap pointT(k+1) in the line section for negative sequence component i=2, or forincremental positive sequence component i=Δ1I_(Fi)—total fault current for negative sequence component i=2, or forincremental positive sequence component i=Δ1,Iki—current measured in station k (the first lower index) for individualsymmetrical components, (the second lower index) i.e. negativecomponent—index 2 and incremental positive sequence component—index Δ1,I_((k+1)i)—current measured in station k+1 (the first lower index) forindividual symmetrical components, (the second lower index) i.e.negative component—index 2 and incremental positive sequencecomponent—index Δ1

ADVANTAGES OF THE INVENTION

The advantage of the method for fault location in electric power lineswhich is the subject of this invention is that it makes it possible todetermine the fault point for a transmission or distribution system bothwith passive and active tap (taps). Due to the required input signals,the inventive location method can be applied in differential-currentprotection, which will increase the functionality of the protectiverelay. In this way, the protective relay, besides its main feature, i.e.indication whether the fault occurred in the given protection zone oroutside it, will be able to exactly define the location of the fault.

In addition, the inventive method is resistant to pre-fault conditionsdefined by the direction and volume of flow of the pre-fault power.

AN EXAMPLE OF THE INVENTION EMBODIMENT

The method according to the present invention is explained in anembodiment shown in the drawing, where

FIG. 1 shows a general diagram of the electric network for theimplementation of the inventive method for a three-terminal electricpower line, with indicated sections LA, LB and LC,

FIG. 2—an equivalent circuit diagram for the positive sequence componentfor the assumption that the fault occurred in line section LA,

FIG. 3—an equivalent circuit diagram of a transmission system for thenegative sequence component for the assumption that the fault occurredin line section LA,

FIG. 4—an equivalent circuit diagram of a transmission system for thezero sequence component for the assumption that the fault occurred inline section LA,

FIG. 5—an equivalent circuit diagram for the positive sequence componentfor the assumption that the fault occurred in line section LB,

FIG. 6—an equivalent circuit diagram of a transmission system for thenegative sequence component for the assumption that the fault occurredin line section LB,

FIG. 7—an equivalent circuit diagram of a transmission system for thezero sequence component for the assumption that the fault occurred inline section LB,

FIG. 8—an equivalent circuit diagram for the positive sequence componentfor the assumption that the fault occurred in line section LC,

FIG. 9—an equivalent circuit diagram of a transmission system for thenegative sequence component for the assumption that the fault occurredin line section LC,

FIG. 10—an equivalent circuit diagram of a transmission system for thezero sequence component for the assumption that the fault occurred inline section LC, and

FIG. 11—an equivalent circuit diagram of a transmission system for thenegative sequence component for the assumption that the fault occurredin line section LA for calculating the impedance of equivalent systems,

FIG. 12—an equivalent circuit diagram for the incremental positivesequence component for the assumption that the fault occurred in linesection LA for calculating the impedance of equivalent systems,

FIG. 13—shows the network of actions performed when locating faults onthe basis of the inventive method for a three-terminal electric powerline,

FIG. 14—a general diagram of the transmission system for theimplementation of the inventive method for a multi-terminal power line,

FIG. 15—an equivalent circuit diagram of a transmission system forsymmetrical components assuming that the fault is located in the firstsection of a multi-terminal power line,

FIG. 16—an equivalent circuit diagram of a transmission system forsymmetrical components for an assumption that the fault is located inthe final section of a multi-terminal power line,

FIG. 17—a fragment of the equivalent circuit diagram of a transmissionsystem for symmetrical components assuming that the fault is located ina section of a tapped line of a multi-terminal power line,

FIG. 18—a fragment of the equivalent circuit diagram of a transmissionsystem for symmetrical components for the assumption that the fault islocated in a section of a multi-terminal power line between two tappoints,

FIG. 19 shows the network of actions performed when locating faults onthe basis of the inventive method for a multi-terminal electric powerline.

AN EXAMPLE OF THE INVENTION EMBODIMENT FOR A THREE-TERMINAL POWER LINE

The transmission system shown in FIG. 1 consists of three electric powerstations A, B and C. Station A is located at the beginning of the line,station B at the end of this line and station C after the line which intap point T is branched out from the line between stations AB. Tap pointT divides the transmission system into three sections LA, LB and LC. Instation A there is a fault locator FL. Fault location is done usingmodels of faults and fault loops for symmetrical components anddifferent types of faults, by applying suitable share coefficientsdetermining the relation between the symmetrical components of the totalfault current when voltage drop across the fault resistance isestimated, defined as a_(F1), a_(F2), a_(F0) and weight coefficients a₁,a₂, a₀, defining the share of individual components in the total modelof the fault loop. Analysis of boundary conditions for different typesof faults shows that there is a certain degree of freedom whendetermining the share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated. Their selection depends on theadopted preference of the use of individual sequence componentsdepending on the type of the fault. In the presented example of theinvention embodiment, in order to ensure high precision of the faultlocation, voltage drop across the fault resistance is estimated using:

the negative sequence component of the total fault current forphase-to-earth faults (a-g), (b-g), (c-g) and phase-to-phase faults(a-b), (b-c) i (c-a),

the negative sequence component and the zero sequence component fordouble phase-to-earth faults (a-b-g), (b-c-g), (c-a-g),

incremental positive sequence component for three-phase faults (a-b-c,a-b-c-g), for which the fault value is decreased by the pre-fault valueof the positive-sequence component of current.

Examples of share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated are shown in table 1. The typeof the fault is denoted by symbols: a-g, b-g, c-g, a-b, b-c, c-a, whereletters a, b, c denote individual phases, and letter g denotes earthing,index 1 denotes the positive-sequence component, index 2—the negativecomponent, and index 0—the zero sequence component.

TABLE 1 Fault (F) a_(F1) a_(F2) a_(F0) a-g 0 3 0 b-g 0  3a 0 c-g 0  3a²0 a-b 0 1 − a 0 b-c 0  a − a² 0 c-a 0 a² − 1  0 a = exp(j2π/3); j ={square root over (−1)}

Synchronised measurements of phase currents from stations A, B, C and ofphase voltages from station A are supplied to the fault locator FL.Additionally, it is assumed that the fault locator is supplied withinformation about the type of the fault and the time of its occurrence.The process of fault location, assuming that it is a fault of the type(a-b-g)—double phase-to-earth fault, is as follows:

I. Stage One

1. In stations A, B, C, current input signals from individual lines forfault and pre-fault conditions are measured. In station A, phasevoltages of the line in fault and pre-fault conditions are measured.Next, the symmetrical components of the phase currents measured instations A, B, C and of phase voltages measured in station A arecalculated.

2. Total fault current (I_(F)) is calculated from this equation:

I _(F) =a _(F1) I _(F1) +a _(F2) I _(F2) +a _(F0) I _(F0)  (1)

where:the first lower index “F” denotes a fault condition, the second lowerindex “1” denotes the positive sequence component, “2”—the negativecomponent, “0”—zero sequence component,and the share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated are as follows:

${{\underset{\_}{a}}_{F\; 1} = 0},{{\underset{\_}{a}}_{F\; 2} = {{\underset{\_}{a}}_{F\; 2}^{{init}.} - \frac{{\underset{\_}{a}}_{F\; 1}^{{init}.}{\underset{\_}{b}}_{F\; 2}}{{\underset{\_}{b}}_{F\; 1}}}},{{\underset{\_}{a}}_{F\; 0} = \frac{{\underset{\_}{a}}_{F\; 1}^{init}}{{\underset{\_}{b}}_{F\; 1}}},$

whereas, in particular for a-b-g fault:

$\begin{matrix}{{{\underset{\_}{a}}_{F\; 1}^{{init}.} = {1 - {\underset{\_}{a}}^{2}}},{{\underset{\_}{a}}_{F\; 2}^{{init}.} = {1 - \underset{\_}{a}}},{{\underset{\_}{b}}_{F\; 1} = {- \underset{\_}{a}}},{{\underset{\_}{b}}_{F\; 2} = {- {\underset{\_}{a}}^{2}}},{\underset{\_}{a} = {\exp^{{j2\pi}/3} = {- 0}}},{5 + {j\frac{\sqrt{3}}{2}}},} & \; \\{{{\underset{\_}{I}}_{F\; 1} = {{\underset{\_}{I}}_{A\; 1} + {\underset{\_}{I}}_{B\; 1} + {\underset{\_}{I}}_{C\; 1}}},} & (2) \\{{{\underset{\_}{I}}_{F\; 2} = {{\underset{\_}{I}}_{A\; 2} + {\underset{\_}{I}}_{B\; 2} + {\underset{\_}{I}}_{C\; 2}}},} & (3) \\{{{\underset{\_}{I}}_{F\; 0} = {{\underset{\_}{I}}_{A\; 0} + {\underset{\_}{I}}_{B\; 0} + {\underset{\_}{I}}_{C\; 0}}},} & (4)\end{matrix}$

where: the first lower index denotes the station, the second lower indexdenotes: 1—the positive sequence component, 2—the negative component,0—the zero sequence component.

For faults of other types, the share coefficients determining therelation between the symmetrical components of the total fault currentwhen voltage drop across the fault resistance is estimated and “relationcoefficients” are compiled in tables 1, 2 and 3.

TABLE 2 Initial share coefficient determining the relation between thesymmetrical components of the total fault current when voltage dropRelation across the fault resistance is estimated coefficient Fault (F)a_(F1) ^(init.) a_(F2) ^(init.) a_(F0) ^(init.) b_(F1) b_(F2) a-b-g  1 −a² 1 − a 0 −a  −a² b-c-g a² − a   a − a² 0 −1 −1 c-a-g a − 1 a² − 1  0 −a² −a

TABLE 3 share coefficient determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated Fault (F) a_(F1) a_(F2) ⁾*a_(F0) a-b-c, a-b-c-g 1 − a² 1 − a 0 ⁾* due to the lack of the negativecomponent this coefficient may be adopted as = 0

II. Stage Two

In stage two, a hypothetical fault point is assumed and the distancebetween the end of the given line section and the hypothetical faultpoint is calculated on the following assumptions:

-   -   calculation of the distance from the beginning of the line to        the fault point assuming that the fault occurred in line section        LA—actions 3.1.a-3.2.a,    -   calculation of the distance from the end of the line to the        fault point assuming that the fault occurred in line section        LB—actions 3.1.b-3.4.b,    -   calculation of the distance from the end of the tapped line to        the fault point assuming that the fault occurred in line section        LC—actions 3.1.c-3.3.c.

3.1.a. The fault loop voltage and current is determined from thefollowing relations between symmetrical components (FIG. 2-4):

$\begin{matrix}{{\underset{\_}{V}}_{Ap} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{A\; 1}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{A\; 2}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{A\; 0}}}} & (5) \\{{{\underset{\_}{I}}_{Ap} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{A\; 1}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{A\; 2}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0{LA}}}{{\underset{\_}{Z}}_{1{LA}}}{\underset{\_}{I}}_{A\; 0}}}},} & (6)\end{matrix}$

where:V_(A1), V_(A2), V_(A0)—voltage measured in station A for individualsymmetrical components, the positive sequence component—index 1, thenegative component—index 2 and the zero sequence component—index 0.I_(A1), I_(A2), I_(A0)—currents measured in station A for the positivesequence component—index 1, the negative component—index 2 and the zerosequence component—index 0,Z_(1LA)—the impedance of line section LA for the positive sequencecomponent,Z_(0LA)—the impedance of line section LA for the zero sequencecomponent, In particular for a-b-g fault, the weight coefficients are:

a ₁=1−a ²,

a ₂=1−a,

a₀=0.

Weight coefficients for other types of faults are compiled in table 4.

TABLE 4 Fault a₁ a₂ a₀ a-g 1 1 1 b-g  a² a 1 c-g a  a² 1 a-b, a-b-g  1 −a² 1 − a 0 a-b-c, a-b-c-g b-c, b-c-g a² − a   a − a² 0 c-a, c-a-g a − 1a² − 1  0

Fault loop equation has the following form:

V _(Ap) −d _(A) Z _(1LA) I _(Ap) −R _(FA) I _(F)=0.  (7)

When the equation (7) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, the solutions shown under 3.2a are obtained.

3.2a. The distance to the fault point d_(A) and the fault resistanceR_(FA) are determined from the following equations:

$\begin{matrix}{{d_{A} = \frac{{{{real}\left( {\underset{\_}{V}}_{Ap} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {\underset{\_}{V}}_{Ap} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}{{{{real}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},} & (8) \\{{R_{FA} = {\frac{1}{2}\left\lbrack {\frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Ap} \right)} -} \\{d_{A}{{real}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{A_{p}}} \right)}}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} + \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Ap} \right)} -} \\{d_{A}{{imag}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} \right\rbrack}},} & (9)\end{matrix}$

where:“real” denotes the real part of the given quantity,“imag” denotes the imaginary part of the given quantity,V_(Ap)—denotes the voltage of the fault loop as per the formula (5)I_(F)—denotes the total fault current as per the formula (1),Z_(1LA)—denotes the impedance of the line section LA for the positivesequence component,I_(Ap)—denotes the fault loop current determined as per the formula (6).

3.1.b. Voltages for symmetrical components are calculated V_(T1)^(transf.), V_(T2) ^(transf.), V_(T0) ^(transf.) (FIG. 5-7) in tap pointT:

V _(T1) ^(transf.)=Cos h(γ_(1LA) l _(LA))·V _(A1) −Z _(c1LA) Sinh(γ_(1LA) l _(LA))·I _(A1),

V _(T2) ^(transf.)=cos h(γ_(1LA) l _(LA))·V _(A2) Z _(c1LA) sinh(γ_(1LA) l _(LA))·I _(A2),

V _(T0) ^(transf.)=cos h(γ_(0LA) l _(LA))·V _(A0) −Z _(c0LA) sinh(γ_(0LA) l _(LA))·I _(A0),

where:γ_(1LA)—propagation constant of line section LA for the positive andnegative sequence components,γ_(0LA)—propagation constant of line section LA for the zero sequencecomponent,l_(LA)—length of line section LA,Z_(c1LA)—surge impedance of section LA for the positive and negativesequence components,Z_(c0LA)—surge impedance of section LA for the zero sequence component.

3.2.b. The values of currents incoming to tap point T from line sectionLA: I_(A1) ^(transf.), I_(A2) ^(transf.), I_(A0) ^(transf.) and sectionLC: I_(C1) ^(transf.), I_(C2) ^(transf.), I_(C0) ^(transf.) (FIG. 5-7)

I _(A1) ^(transf.)=(−1/Z _(c1LA))·sin h(γ_(1LA) l _(LA))·V _(A1)+cosh(γ_(1LA) l _(LA))·I _(A1),

I _(A2) ^(transf.)=(−1/Z _(c1LA))·sin h(γ_(1LA) l _(LA))·V _(A2)+cosh(γ_(1LA) l _(LA))·I _(A2),

I _(A0) ^(transf.)=(−1/Z _(c0LA))·sin h(γ_(0LA) l _(LA))·V _(A0)+cosh(γ_(0LA) l _(LA))·I _(A0),

I _(C1) ^(transf.)=(−1/Z _(c1LC))·tan h(γ_(1LC) l _(LC))·V _(T1)^(transf.)+(1/cos h(γ_(1LC) l _(LC)))·I _(C1),

I _(C2) ^(transf.)=(−1/Z _(c1LC))·tan h(γ_(1LC) l _(LC))·V _(T2)^(transf.)+(1/cos h(γ_(1LC) l _(LC)))·I _(C2),

I _(C0) ^(transf.)=(−1/Z _(c0) LC)·tan h(γ_(0LC) l _(LC))·V _(T0)^(transf.)+(1/cos h(γ_(0LC) l _(LC)))·I _(C0),

where:γ_(1LC)—propagation constant of line section LC for the positive andnegative sequence components,γ_(0LC)—propagation constant of line section LC for the zero sequencecomponent,Z_(c1LC)—surge impedance of section LC for the positive and negativesequence components,Z_(c0LC)—surge impedance of section LC for the zero sequence component,l_(LC)—length of line section LC.

3.3.b. The values of current I_(TB1) ^(transf.), I_(TB2) ^(transf.),I_(TB0) ^(transf.) flowing from tap point T to station B in line sectionLB is calculated:

I _(TB1) ^(transf.) =I _(A1) ^(transf.) +I _(C1) ^(transf.),

I _(TB2) ^(transf.) =I _(A2) ^(transf.) +I _(C2) ^(transf.),

I _(TB0) ^(transf.) =I _(A0) ^(transf.) +I _(C0) ^(transf.).

Fault loop equation has the following form:

V _(Tp)−(1−d _(B)) Z_(1LB) I _(TBp) −R _(FB) I _(F)=0,  (10)

where:

V _(Tp) =a ₁ V _(T1) ^(transf.) +a ₂ V _(T2) ^(transf.) +a ₀ V _(T0)^(transf.)

I _(TBp) =a ₁ I _(TB1) ^(transf.) +a ₂ I _(TB2) ^(transf.) +a ₀ Z _(0LB)/Z _(1LB) I _(TB0) ^(transf.)  (11)

When the equation (10) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, the solutions shown under 3.4b are obtained.

3.4.b. The distance to fault point d_(B) and fault resistance R_(FB) arecalculated from the following equations:

$\begin{matrix}{{d_{B} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{{{{real}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},} & \left. 12 \right) \\{{R_{FB} = {{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{B}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} + {\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{B}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}},} & (13)\end{matrix}$

where:Z_(1LB)—impedance of line section LB for the positive sequencecomponent,Z_(0LB)—impedance of line section LB for the zero sequence component.

3.1.c. The values of currents incoming to tap point T (FIG. 8-10) fromline sections LA I_(A1) ^(transf.), I_(A2) ^(transf.), I_(A0) ^(transf.)and section LB I_(B1) ^(transf.), I_(B2) ^(transf.)I_(B0) ^(transf.) arecalculated according to these formulas:

I _(B1) ^(transf.)=(−1/Z _(c1) LB)·tan h(γ_(1LB) l _(LB))·V _(T1)^(transf.)+(1/cos h(γ_(1LB) l _(LB)))·I _(B1),

I _(B2) ^(transf.)=(−1/Z _(c1) LB)·tan h(γ_(1LB) l _(LB))·V _(T2)^(transf.)+(1/cos h(γ_(1LB) l _(LB)))·I _(B2),

I _(B0) ^(transf.)=(−1/Z _(c0) LB)·tan h(γ_(0LB) l _(LB))·V _(T0)^(transf.)+(1/cos h(γ_(0LB) l _(LB)))·I _(B0),

I _(A1) ^(transf.)=(−1/Z _(c1LA))·sin h(γ_(1LA) l _(LA))·V _(A1)+cosh(γ_(1LA) l _(LA))·I _(A1),

I _(A2) ^(transf.)=(−1/Z _(c1LA))·sin h(γ_(1LA) l _(LA))·V _(A2)+cosh(γ_(1LA) l _(LA))·I _(A2),

I _(A0) ^(transf.)=(−1/Z _(c0LA))·sin h(γ_(0LA) l _(LA))·V _(A0)+cosh(γ_(0LA) l _(LA))·I _(A0),

where:γ_(1LB)—propagation constant of line section LB for the positive andnegative sequence components,γ_(0LB)—propagation constant of line section LB for the zero sequencecomponent,l_(LB)—the length of line section LB.Z_(c1LB)—surge impedance of section LB for the positive sequence andnegative components,Z_(c0LB)—surge impedance of section LB for the zero sequence component.

3.2.c. The value of current I_(TC1) ^(transf.), I_(TC2) ^(transf.),I_(TC0) ^(transf.) flowing from tap point T to station C in line sectionLC (FIG. 8-10) is calculated:

I _(TC1) ^(transf.) =I _(A1) ^(transf.) +I _(B1) ^(transf.),

I _(TC2) ^(transf.) =I _(A2) ^(transf.) +I _(B2) ^(transf.),

I _(TC0) ^(transf.) =I _(A0) ^(transf.) +I _(B0) ^(transf.).

Fault loop equation has the following form:

$\begin{matrix}{{{{\underset{\_}{V}}_{Tp} - {\left( {1 - d_{C}} \right){\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} - {R_{FC}{\underset{\_}{I}}_{F}}} = 0}{{where}:}} & (14) \\{{{\underset{\_}{V}}_{Tp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{T\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{T\; 2}^{{transf}.}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{T\; 0}^{{transf}.}}}}{{\underset{\_}{I}}_{TCp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{{TC}\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{{TC}\; 2}^{{transf}.}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0{LC}}}{{\underset{\_}{Z}}_{1{LC}}}{\underset{\_}{I}}_{{TC}\; 0}^{{transf}.}}}}} & (15)\end{matrix}$

When the equation (13) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, the solutions shown under 3.3.c are obtained.

3.3.c. The distance to fault point d_(C) and fault resistance R_(FC) arecalculated from the following equations:

$\begin{matrix}{{d_{C} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{{{{real}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},} & (16) \\{R_{FC} = {{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{C}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} + {{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{C}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}.}}} & (17)\end{matrix}$

III. Stage Three

At this stage the selection of final results is done.

4. It is checked whether the results of the calculation of the distancesd_(A), d_(B), d_(C) to the fault point are contained within the interval(0÷1) in relative units:

0≦d_(A)≦1,

0≦d_(B)≦1,

0≦d_(C)≦1.

Results which are not contained in the given interval show that theyhave been calculated on a false pre-assumption concerning the point ofoccurrence of the fault on the given line section. These results arerejected.

5. The second calculated value, i.e. fault resistance R_(FA), R_(FB),R_(FC) is analysed and those results of the calculations for which thefault resistance is negative are rejected.

6. If the analysis of the criteria as in actions 4 and 5 does notindicate clearly which values define the place of the fault, then insubsequent actions the impedance of equivalent source systems for thenegative component in the case of phase-to-earth, phase-to-phase, doublephase-to earth faults, or alternatively for the incremental positivesequence component is calculated. For three-phase faults, the impedanceof equivalent source systems for the incremental positive sequencecomponent is calculated.

7. Current I_(FA2) (FIG. 11) is calculated assuming that the faultoccurred in line section LA:

${{\underset{\_}{I}}_{{FA}\; 2} = {\frac{{\underset{\_}{Z}}_{2{SA}} + {\underset{\_}{Z}}_{2{LA}} + {{\underset{\_}{Q}}_{{BC}\; 2}\left( {{\underset{\_}{Z}}_{2{LB}} + {\underset{\_}{Z}}_{2{SB}}} \right)}}{{\left( {1 - d_{A}} \right){\underset{\_}{Z}}_{2{LA}}} + {{\underset{\_}{Q}}_{{BC}\; 2}\left( {{\underset{\_}{Z}}_{2{LB}} + {\underset{\_}{Z}}_{2{SB}}} \right)}}{\underset{\_}{I}}_{A\; 2}}},{{{where}:{\underset{\_}{Q}}_{{BC}\; 2}} = \frac{{\underset{\_}{I}}_{B\; 2}}{{\underset{\_}{I}}_{B\; 2} + {\underset{\_}{I}}_{C\; 2}}},$

8. The equivalent source impedance (Z_(2SB))_(SUB) _(—) _(A) iscalculated assuming that the fault occurred in line section LA:

$\begin{matrix}{{\left( {\underset{\_}{Z}}_{2{SB}} \right)_{SUB\_ A} = \frac{{{\underset{\_}{G}}_{2A}{\underset{\_}{I}}_{A2}} - {{\underset{\_}{H}}_{2A}{\underset{\_}{I}}_{{FA}\; 2}}}{{\underset{\_}{Q}}_{{BC}\; 2}\left( {{\underset{\_}{I}}_{{FA}\; 2} - {\underset{\_}{I}}_{A\; 2}} \right)}},{{{where}:\text{}{\underset{\_}{G}}_{2A}} = {{\underset{\_}{Z}}_{2{SA}} + {\underset{\_}{Z}}_{2{LA}} + {{\underset{\_}{Q}}_{{BC}\; 2}{\underset{\_}{Z}}_{2{LB}}}}},{{\underset{\_}{H}}_{2A} = {{\left( {1 - d_{A}} \right){\underset{\_}{Z}}_{2{LA}}} + {{\underset{\_}{Q}}_{{BC}\; 2}{\underset{\_}{Z}}_{2{LB}}}}},{{\underset{\_}{Z}}_{2{SA}} = {\frac{- {\underset{\_}{V}}_{A\; 2}}{{\underset{\_}{I}}_{A\; 2}}.}},} & (18)\end{matrix}$

9. The equivalent source impedance (Z_(2SC))_(SUB) _(—) _(A) iscalculated assuming that the fault occurred in line section LA:

$\begin{matrix}{\left( {\underset{\_}{Z}}_{2{SC}} \right)_{SUB\_ A} = {{\left( {{\underset{\_}{Z}}_{2{LB}} + \left( {\underset{\_}{Z}}_{2{SB}} \right)_{SUB\_ A}} \right)\frac{{\underset{\_}{I}}_{B\; 2}}{{\underset{\_}{I}}_{C\; 2}}} - {{\underset{\_}{Z}}_{2{LC}}.}}} & (19)\end{matrix}$

10. The equivalent source impedance (Z_(2SB))_(SUB) _(—) _(B) iscalculated assuming that the fault occurred in line section LB:

$\begin{matrix}{\left( {\underset{\_}{Z}}_{2{SB}} \right)_{SUB\_ B} = {\frac{{\left( {1 - d_{B}} \right){\underset{\_}{Z}}_{2{LB}}{\underset{\_}{I}}_{{TB}\; 2}^{{transf}.}} - {d_{B}{\underset{\_}{Z}}_{2{LB}}{\underset{\_}{I}}_{B\; 2}} - {\underset{\_}{V}}_{T\; 2}^{{transf}.}}{{\underset{\_}{I}}_{B\; 2}}.}} & (20)\end{matrix}$

11. The equivalent source impedance (Z_(2SC))_(SUB) _(—) _(B) iscalculated assuming that the fault occurred in line section LB:

$\begin{matrix}{{{\left( {\underset{\_}{Z}}_{2{SC}} \right)_{SUB\_ B} = {- \frac{{\underset{\_}{V}}_{C\; 2}}{{\underset{\_}{I}}_{C\; 2}}}}{{where}:\text{}{\underset{\_}{V}}_{C\; 2}} = {{{\cosh \left( {{\underset{\_}{\gamma}}_{2{LC}}l_{LC}} \right)} \cdot {\underset{\_}{V}}_{T\; 2}^{{transf}.}} + {{\underset{\_}{Z}}_{c\; 2{LC}}{{\sinh \left( {{\underset{\_}{\gamma}}_{2{LC}}l_{LC}} \right)} \cdot {\underset{\_}{I}}_{C\; 2}^{{transf}.}}}}},{{\underset{\_}{\gamma}}_{2{LC}} = {\underset{\_}{\gamma}}_{1{LC}}},{{\underset{\_}{Z}}_{c\; 2{LC}} = {{\underset{\_}{Z}}_{c\; 1{LC}}.}}} & (21)\end{matrix}$

12. The equivalent source impedance (Z_(2SB))_(SUB) _(—) _(c) iscalculated assuming that the fault occurred in line section LC:

$\begin{matrix}{\left( {\underset{\_}{Z}}_{2{SC}} \right)_{SUB\_ C} = {\frac{{\left( {1 - d_{C}} \right){\underset{\_}{Z}}_{2{LC}}{\underset{\_}{I}}_{{TC}\; 2}^{{transf}.}} - {d_{C}{\underset{\_}{Z}}_{2{LC}}{\underset{\_}{I}}_{C\; 2}} - {\underset{\_}{V}}_{T\; 2}^{{transf}.}}{{\underset{\_}{I}}_{C\; 2}}.}} & (22)\end{matrix}$

13. The equivalent source impedance (Z_(2SB))_(SUB) _(—) _(C) iscalculated assuming that the fault occurred in line section LC:

$\begin{matrix}{{{\left( {\underset{\_}{Z}}_{2{SB}} \right)_{SUB\_ C} = {- \frac{{\underset{\_}{V}}_{B\; 2}}{{\underset{\_}{I}}_{B\; 2}}}}{{where}:{\underset{\_}{V}}_{B\; 2}} = {{{\cosh \left( {{\underset{\_}{\gamma}}_{2{LB}}l_{LB}} \right)} \cdot {\underset{\_}{V}}_{T\; 2}^{{transf}.}} + {{\underset{\_}{Z}}_{c\; 2{LB}}{{\sinh \left( {{\underset{\_}{\gamma}}_{2{LB}}l_{LB}} \right)} \cdot {\underset{\_}{I}}_{B\; 2}^{{transf}.}}}}},} & (23)\end{matrix}$

14. The calculated equivalent source impedances are transformed into amodular form, whereupon the proper result is selected on the basis ofthe module of the equivalent source system impedances.

If the calculated value of the module of the equivalent source systemimpedances, assuming the occurrence of the fault on the given linesection, does not correspond to the real value of the source systemimpedance module, it means that the preliminary data concerning theplace of occurrence of the fault on the given section have been assumedwrongly, and the result of the calculation of the distance to the faultpoint made on this assumption is rejected.

If the value of the module of impedance of the equivalent source systemscalculated on the assumption that the fault occurred in the given linesection corresponds to the real value of the module of impedance of theequivalent source system, then the result of the calculation of thedistance to the fault point indicates a correct pre-assumption and thisresult is considered to be final.

The network of actions shown in FIG. 13 includes the following actionsfor the implementation of the invention:

-   -   current and voltage measurement as per point 1 of the example        embodiment of the invention,    -   determination of the symmetrical components of the measured        currents and voltages and calculation of the total fault current        as per point 2 of the invention embodiment example,    -   calculation of three hypothetical distances to the fault points        and three fault resistances assuming that the fault occurred in        section LA, section LB and section LC, as per points:        3.1.a-3.2.a, 3.1.b-3.4.b, 3.1.c-3.3.c of the invention        embodiment example,    -   checking whether the particular hypothetical distances are        contained in the interval from 0 to 1 in relative units and        rejection of those hypothetical distances whose values are        negative or bigger than 1, according to point 4 of the invention        embodiment example.    -   checking whether the fault resistance values are bigger than or        equal to zero and rejection of values less than zero, according        to point 5 of the invention embodiment example,    -   calculation of the impedance of equivalent sources of individual        sections assuming that the fault occurred in the given section,        as per points 8-13 of the invention embodiment example,    -   selection of the correct result, as per point 14 of the        invention embodiment example.

The described example applies to a double phase-to-earth fault of thetype (a-b-g). Yet this method is analogous for other types of faults. Ifother types of faults are analysed, the relevant coefficients a_(F1),a_(F2), a_(F0), a₁, a₂, a₀ change. The values of these coefficients arecompiled in tables 1-4. The method for fault location in three-terminalelectric power transmission lines according to the present inventioncovers also other types of faults, i.e. (a-g, b-g, c-g, a-b, b-c, c-a,b-c-g, c-a-g, a-b-c, a-b-c-g).

The inventive method is not restricted to one line model presented inthe example of the analysis, but it can apply to another model, notshown in the figure, in which model the presence of series compensatingcapacitors in the line section with a fault is assumed. In such case theequations (6), (11), (15) applicable to fault loop current will bemodified due to the existence of these capacitors.

The inventive method uses synchronous measurements of currents in threestations of the transmission or distribution system, additionally itemploys voltage measurement in the station where the fault locator isinstalled. Such availability of input signals is not considered in othersolutions that are currently in use.

The selection of the valid result is based on the aggregation of threecriterion quantities: distance to the fault point, fault resistance inthe fault point and the module of impedance of equivalent source systemsfor those stations where voltage is not measured. This third criterionis innovative and has not been known till now.

An Example of the Invention Embodiment for a Multi-Terminal Power Line.

The transmission or distribution system shown in FIG. 14 consists of 1,2, . . . , n electric power stations. Station 1 is at the beginning ofthe line, the n^(th) station is at the end of the line. Tap points T1,T2, . . . T(n−1) divide the transmission system into line sections L1,L2, . . . , L(2n−3). In station 1 there is a fault locator FL. Faultlocation is done using models of faults and fault loops for symmetricalcomponents and taking into consideration different types of faults atthe same time, by applying suitable share coefficients determining therelation between the symmetrical components of the total fault currentwhen voltage drop across the fault resistance is estimated, defined asa_(F1), a_(F2), a_(F0) and weight coefficients a₁, a₂, a₀ defining theshare of individual components in the total model of a fault loop. Theanalysis of boundary conditions for different types of faults indicatesthe existence of a certain degree of freedom at determining sharecoefficients determining the relation between the symmetrical componentsof the total fault current when voltage drop across the fault resistanceis estimated. Their selection depends on the adopted preference in theuse of individual components depending on the type of the fault. In thepresented example of the invention embodiment, in order to ensure highprecision of fault location, voltage drop across the fault resistance isestimated using:

-   -   the negative component of the total fault current for        phase-to-earth faults (a-g), (b-g), (c-g) and phase-to-phase        faults (a-b), (b-c) and (c-a),    -   the negative component and the zero sequence component for        double phase-to-earth faults (a-b-g), (b-c-g), (c-a-g),    -   incremental positive-sequence component for three-phase faults        (a-b-c, a-b-c-g) for which the fault value is decreased by the        pre-fault value of the positive sequence component of current.

The recommended share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated are presented in table 5. Thetype of the fault is denoted by symbols: a-g, b-g, c-g, a-b, b-c, c-a,a-b-g, b-c-g, c-a-g, a-b-c, a-b-c-g where letters a, b, c denoteindividual phases of the current, and letter g denotes earthing(ground), index 1 denotes the positive-sequence component, index 2—thenegative sequence component, and index 0—the zero sequence component.

TABLE 5 Fault a_(F1) a_(F2) a_(F0) a-g 0 3 0 b-g 0 1.5 + j1.5{squareroot over (3)} 0 c-g 0 −1.5 − j1.5{square root over (3)}  0 a-b 0 1.5 −j1.5{square root over (3)} 0 b-c 0 j{square root over (3)} 0 c-a 0 −1.5− j0.5{square root over (3)}  0 a-b-g 0 3 − j{square root over (3)}  j{square root over (3)} b-c-g 0 j2{square root over (3)} j{square rootover (3)} c-a-g 0 −3 − j{square root over (3)}   j{square root over (3)}a-b-c 1.5 + j0.5{square root over (3)}  1.5 − j0.5{square root over (3)}⁾* 0 a-b-c-g ⁾* due to the lack of the negative component thiscoefficient may be adopted as = 0

In table 6 there are compiled share coefficients of individual currentcomponents a₁, a₂, a₀, defining the share of individual components inthe total model of the fault loop.

TABLE 6 FAULT a₁ a₂ a₀ a-g 1 1 1 b-g −0.5 − j0.5{square root over (3)} 0.5 + j0.5{square root over (3)} 1 c-g 0.5 + j0.5{square root over (3)}−0.5 − j0.5{square root over (3)}  1 a-b, a-b-g 1.5 + j0.5{square rootover (3)} 1.5 − j0.5{square root over (3)} 0 a-b-c, a-b-c-g b-c, b-c-g−j{square root over (3)} j{square root over (3)} 0 c-a, c-a-g −1.5 +j0.5{square root over (3)}  −1.5 − j0.5{square root over (3)}  0

Synchronised measurements of phase currents from all terminal stationsof lines 1, 2, . . . , n and phase voltages only from station 1 aresupplied to the fault locator FL. Additionally, it is assumed thatinformation about the type of the fault and the time of its occurrenceis supplied to the fault locator. The fault location process assumingthat it is a fault of the type (a-b-g)—phase-to-phase-to earth fault isas follows:

I′. Stage One′

1′. Operation 610. Input current signals from individual lines for faultand pre-fault conditions are measured in stations 1, 2, . . . , n. Phasevoltages of the line for fault and pre-fault conditions are measured instation 1. Next, symmetrical components of phase currents measured instations 1, 2, . . . , n, and of phase voltages measured in station 1are calculated.

2′. Operation 620. Total fault current (I_(F)) is calculated from thisequation:

I _(F) =a _(F1) I _(F1) +a _(F2) I _(F2) +a _(F0) I _(F0)  (24)

where:the first lower index “F” denotes the fault condition, the second lowerindex “1” denotes the positive sequence component, “2”—the negativesequence component, “0”—the zero sequence component,a_(F1), a_(F2), a_(F0)—the coefficients presented in table 2,

The symmetrical components of total fault current are determined as thesum of individual symmetrical components of currents determined in allterminal stations 1, 2, . . . , n:

I _(F1) =I ₁₁ +I ₂₁ +I ₃₁+ . . . +I_((n−1)1) +I _(n1,)  (25)

I _(F2) =I ₁₂ +I ₂₂ +I ₃₂+ . . . +I_((n−1)2) +I _(n2,)  (26)

I _(F0) =I ₁₀ +I ₂₀ +I ₃₀+ . . . +I_((n−1)0) +I _(n0.)  (27)

where: the first lower index denotes the station, the second lower indexdenotes: 1—the positive sequence component, 2—the negative component,0—the zero sequence component.

II′. Stage Two′

In stage two′ a hypothetical fault point is assumed and the distancebetween the end of the given line and the hypothetical fault point iscalculated (actions performed in operations 630 a, 630 b, 630 c, 630 d),on the following assumptions:

-   -   calculation of the distance from the beginning of the line to        the fault point assuming that the fault occurred in the first        line section L1—actions 3.1.a′    -   calculation of the distance from the end of the line to the        fault point assuming that the fault occurred in the terminal        line section L(2n−3)—actions 3.1.b′-3.3.b′.    -   calculation of the distance from the end of the tapped line to        the fault point assuming that the fault is located in the k^(th)        tapped line—actions 3.1.c′-3.2.c′,    -   calculation of the distance from tap point T(k) to the fault        point assuming that the fault is located in the line section        between two tap points—actions 3.1.d′-3.3d′.

3.1.a′. The voltage and current of the fault loop is determined from therelation between the symmetrical components (FIG. 2-4), (actionsperformed in operations 630 a):

$\begin{matrix}{{{\underset{\_}{V}}_{1p} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{11}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{12}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{10}}}},} & (28) \\{{\underset{\_}{I}}_{1p} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{11}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{12}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0L\; 1}}{{\underset{\_}{Z}}_{1L\; 1}}{\underset{\_}{I}}_{10}}}} & (29)\end{matrix}$

where:V₁₁, V₁₂, V₁₀—voltage measured in station 1 (the first lower index) forindividual symmetrical components, (the second lower index) i.e. thepositive sequence component—index 1, negative component—index 2 and zerosequence component—index 0.I₁₁, I₁₂, I₁₀—currents measured in station 1 (the first lower index) forindividual symmetrical components, (the second lower index) i.e. thepositive sequence component—index 1, negative component—index 2 and zerosequence component—index 0.Z_(1L1)—impedance of line section L1 for the positive sequencecomponent,Z_(0L1)—impedance of line section L1 for the zero sequence component,a₁, a₂, a₀,—weight coefficients compiled in table 2.

Fault loop equation has the following form:

V _(1p) −d ₁ Z _(1L1) I _(1p) −R _(1F) I _(F)=0  (30)

When the equation (30) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, these equations for the searched distance to the faultpoint (31) and fault resistance (32) are obtained.

$\begin{matrix}{{d_{1} = \frac{{{{real}\left( {\underset{\_}{V}}_{1p} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {\underset{\_}{V}}_{1p} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}{{{{real}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},} & (31) \\{{R_{1F} = {\frac{1}{2}\left\lbrack {\frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{1p} \right)} -} \\{d_{1}{{real}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} + \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{1p} \right)} -} \\{d_{1}{{imag}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} \right\rbrack}},} & (32)\end{matrix}$

where:“real” denotes the real part of the given complex quantity,“imag” denotes the imaginary part of the given complex quantity,V_(1p)—denotes the voltage of the fault loop as per the formula (28)I_(F)—denotes the total fault current as per the formula (25),Z_(1L1)—denotes the impedance of the line section L1 for the positivesequence component,I_(1p)—denotes the fault loop current determined as per the formula(29).

3.1.b′. Voltages for symmetrical components V_(T11) ^(transf.), V_(T12)^(transf.), V_(T10) ^(transf.) in the first tap point T2 are calculated,(actions performed in operations 630 b):

V _(T2) i ^(transf.) =V _(1i) −Z _(iL1) ·I _(1i),  (33)

where:Z_(iL1)—impedance of section L1 respectively for the positive andnegative components and for the zero sequence component.

3.2.b′ Voltages for symmetrical components V_(T(n−1)1) ^(transf.),V_(T(n−1)2) ^(transf.), V_(T(n−4)0) ^(transf.) in the final tap pointT(n−1) are calculated (actions performed in operations 630 b):

$\begin{matrix}{{{\underset{\_}{V}}_{{T{({n - 1})}}i}^{{transf}.} = {{\underset{\_}{V}}_{{T{({n - 2})}}i}^{{transf}.} - {{\underset{\_}{Z}}_{{iL}{({{2n} - 5})}} \cdot {\sum\limits_{\underset{{i = 1},2,3}{j = 1}}^{n - 2}{\underset{\_}{I}}_{ji}}}}},} & (34)\end{matrix}$

where:Z_(iL(2n−3))—impedance of line section L(2n−3) respectively for thepositive and negative components and for the zero sequence component.

At the same time voltages in the k^(th) tap point V_(Tk1) ^(transf.),V_(Tk2) ^(transf.), V_(Tk0) ^(transf.) are determined from the followingformula:

$\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{transf}.} = {{\underset{\_}{V}}_{{T{({k - 1})}}i}^{{transf}.} - {{\underset{\_}{Z}}_{{iL}{({{2k} - 3})}} \cdot {\sum\limits_{\underset{{i = 1},2,3}{j = 1}}^{k - 1}{\underset{\_}{I}}_{ji}}}}} & (35)\end{matrix}$

where:V_(T(k−1)i) ^(transf.)—calculated voltage in point (k−1),Z_(iL(2k−3))—impedance of line section L(2k−3) for the symmetricalcomponents.

3.3.b′. The values of current I_(T(n−1)n1) ^(transf.), I_(T(n−1)ns2)^(transf.), I_(T(n−1)n0) ^(transf.) flowing from tap point T(n−1) tostation n in line section L(2n−3) are calculated, (actions performed inoperations 630 b):

$\begin{matrix}{{{\underset{\_}{I}}_{{T{({n - 1})}}{ni}}^{{transf}.} = {\sum\limits_{\underset{j = 1}{{i = 1},2,0}}^{n - 1}{\underset{\_}{I}}_{ji}}},} & (36)\end{matrix}$

Fault loop equation has the following form:

$\begin{matrix}{{{{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {\left( {1 - d_{({{2n} - 3})}} \right){\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} - {R_{{({{2n} - 3})}F}{\underset{\_}{I}}_{F}}} = 0}{{where}\text{:}}} & (37) \\{{\underset{\_}{V}}_{{T{({n - 1})}}{np}} = {{{\underset{\_}{a}}_{1}V_{{T{({n - 1})}}1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{{T{({n - 1})}}2}^{{transf}.}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{{T{({n - 1})}}0}^{{transf}.}}}} & \left( {37a} \right) \\{{\underset{\_}{I}}_{{T{({n - 1})}}{np}} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{{T{({{2n} - 1})}}n\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{{T{({n - 1})}}n\; 2}^{{transf}.}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0{L{({{2n} - 3})}}}}{{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}}{\underset{\_}{I}}_{{T{({n - 1})}}n\; 0}^{{transf}.}}}} & \left( {37b} \right)\end{matrix}$

When the equation (37) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, solutions for the searched distance to the fault pointd_((2n−3)) (38) and fault resistance R_((2n−3)B) (39) are obtained:

$\begin{matrix}{\mspace{79mu} {{d_{({{2n} - 3})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}},}} & (38) \\{{R_{{({{2n} - 3})}F} = {{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{{T{({n - 1})}}{np}} \right)} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}++}{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{{T{({n - 1})}}{np}} \right)} - {\left( {1 - d_{({{3n} - 3})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{({{2n} - 3})}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}},} & (39)\end{matrix}$

where:Z_(1L(2n−3))—impedance of line section L(2n−3) for the positive sequencecomponent,Z_(0L(2n−3))—impedance of line section L(2n−3) for the zero sequencecomponent.

3.1.c′. Voltages for symmetrical components V_(Tk1) ^(transf.), V_(Tk2)^(transf.), V_(Tk0) ^(transf.) in the 0 tap point Tk are calculated fromthe formula (35) assuming as k the number of the considered stationequal to the number of the tap point Tk from which the line in which weconsider the fault goes to station k (actions performed in operations630 c).

3.2.c′. The values of current I_(Tkk1) ^(transf.), I_(Tkk2) ^(transf.),I_(Tkk0) ^(transf.) flowing from tap point Tk to k^(th) station intapped line section L(2k−2) are calculated (actions performed inoperations 630 c):

$\begin{matrix}{{\underset{\_}{I}}_{Tkki}^{transf} = {\sum\limits_{\underset{{j = 1},{j \neq k}}{{i = 1},2,3}}^{n}{\underset{\_}{I}}_{{ji},}}} & (40)\end{matrix}$

Fault loop equation has the following form:

$\begin{matrix}{{{{\underset{\_}{V}}_{Tkkp} - {\left( {1 - d_{({{2k} - 2})}} \right){\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} - {R_{{({{2k} - 2})}k}{\underset{\_}{I}}_{F}}} = 0}{{where}\text{:}}} & (41) \\{{\underset{\_}{V}}_{Tkkp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{{Tkk}\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{{Tkk}\; 2}^{{transf}.}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{{Tkk}\; 0}^{{transf}.}}}} & \left( {41a} \right) \\{{\underset{\_}{I}}_{Tkkp} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{{Tkk}\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{{Tkk}\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0\; {L{({{2k} - 2})}}}}{{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}}{\underset{\_}{I}}_{{Tkk}\; 0}^{{transf}.}}}} & \left( {41b} \right)\end{matrix}$

When the equation (40) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, solutions for the searched distance to the fault pointd_((2k−2)) (42) and fault resistance R_((2k−2)F) (43) are obtained:

$\begin{matrix}{\mspace{79mu} {{d_{({{2k} - 2})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tkkp} - {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{Tkkp} - {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}},}} & (42) \\{R_{{({{2k} - 2})}F} = {{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tkkp} \right)} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}++}{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tkkp} \right)} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}} & (43)\end{matrix}$

where:Z_(1L(2k−2))—impedance of line section L(2k−2) for the positive sequencecomponent,Z_(0L(2k−2))—impedance of line section L(2k−2) for the zero sequencecomponent.

3.1.d′. Voltages for symmetrical components V_(Tk1) ^(transf.), V_(Tk2)^(transf.), V_(Tk0) ^(transf.) in the k^(th) tap point Tk are calculatedfrom the formula (35), (actions performed in operations 630 d).

3.2.d′. The values of current I_(TkT(k+1)1) ^(transf.), I_(TkT(k+1)2)^(transf.), I_(TkT(k+1)0) ^(transf.) flowing from tap point Tk to tappoint T(k+1) in the line section are calculated (actions performed inoperations 630 d):

$\begin{matrix}{{I_{{{TkT}{({k + 1})}}i}^{{transf}.} = {\sum\limits_{\underset{j = 1}{{i = 1},2,0}}^{k - 1}I_{ji}}},} & (44)\end{matrix}$

Fault loop equation has the following form:

$\begin{matrix}{\mspace{79mu} {{{{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {\left( {1 - d_{({k + 1})}} \right){\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} - {R_{{({{2k} - 1})}E}{\underset{\_}{I}}_{F}}} = 0}\mspace{79mu} {{where}\text{:}}}} & (45) \\{\mspace{79mu} {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} = {{{\underset{\_}{a}}_{1}{\underset{\_}{V}}_{{Tk}\; 1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{V}}_{{Tk}\; 2}^{{transf}.}} + {{\underset{\_}{a}}_{0}{\underset{\_}{V}}_{{Tk}\; 0}^{{transf}.}}}}} & \left( {45a} \right) \\{{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p} = {{{\underset{\_}{a}}_{1}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}1}^{{transf}.}} + {{\underset{\_}{a}}_{2}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}2}^{{transf}.}} + {{\underset{\_}{a}}_{0}\frac{{\underset{\_}{Z}}_{0{L{({k + 1})}}}}{{\underset{\_}{Z}}_{1{L{({k + 1})}}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}0}^{{transf}.}}}} & \left( {45b} \right)\end{matrix}$

When the equation (45) has been written out separately for the real partand the imaginary part and further mathematical transformations havebeen performed, equations for the searched distance to the fault pointd_((2k−1)) (46) and fault resistance R_((2k−1)F) (47) are obtained:

$\begin{matrix}{\mspace{79mu} {{d_{({{2k} - 1})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\\begin{matrix}{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}\end{matrix}}},}} & (46) \\{{R_{{({{2k} - 1})}F} = {{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} \right)} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}++}{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} \right)} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}},} & (47)\end{matrix}$

where:Z_(1L(2k−1))—impedance of line section L(2k−1) for the positive sequencecomponent,Z_(0L(2k−1))—impedance of line section L(2k−1) for the zero sequencecomponent.k—number of the tap point

III′. Stage Three′

In this stage the selection of the final results is done (actionsperformed in operations 640 a, 640 b, 640 c, 640 d).

4′. It is checked whether the results of the calculation of the distanced₁, d_((2n−3)), d_((2k−2)), d_((2k−1)), to the fault point are containedwithin the interval (0÷1) in relative units:

0≦d₁≦1, 0≦d_((2n−3))≦1, 0≦d_((2k−2))≦1, 0≦d_((2k−1))≦1, and it ischecked whether the results of the calculation of fault resistanceR_(1F), R_((2n−3)F), R_((2k−2)F), R_((2k−1)F), for the calculated faultpoints d₁, d_((2n−3)), d_((2k−2)), d_((2k−1)), are bigger than or equalto zero. The pairs of results: resistance-distance, e.g.: d₁, R_(1F)which are not contained within the given intervals indicate that theywere calculated on a false pre-assumption concerning the place ofoccurrence of the fault on the given line section. These results shallbe discarded. The other results undergo further treatment except thecase where only one pair is within the given interval. These results arefinal, i.e. they indicate the fault location and the fault resistance atthe fault point (Operation 650).

5′. If the analysis of the criteria performed as in actions 4′ does notprovide an explicit conclusion about which values define the place andresistance of the fault then, in subsequent actions, equivalent sourceimpedance for the negative component is calculated for these faults:phase-to-earth, phase-to-phase, double phase-to-earth or alternativelyfor the incremental positive sequence component. For three-phase faults,the impedance of equivalent source systems is calculated for theincremental positive sequence component. (Actions performed inoperations 660 a, 660 b, 660 c, 660 d).

7′. Total fault current I_(F2) for the negative sequence component iscalculated from the following formula (actions performed in operations660 a),

$\begin{matrix}{{\underset{\_}{I}}_{F2} = {\sum\limits_{j = 1}^{n}{\underset{\_}{I}}_{j2}}} & (48)\end{matrix}$

8′. Equivalent source impedance (Z_(2S1)) is calculated assuming thatthe fault is located in line section L1, (actions performed inoperations 660 a):

$\begin{matrix}{{\left( {\underset{\_}{Z}}_{2S\; 1} \right) = \frac{- {\underset{\_}{V}}_{12}}{{\underset{\_}{I}}_{12}}},} & (49)\end{matrix}$

9′. Equivalent source impedance (Z_(2S(n))) is calculated assuming thatthe fault is located in the final line section L(2n−3), (actionsperformed in operations 660 b):

$\begin{matrix}{\mspace{79mu} {{\left( {\underset{\_}{Z}}_{2{Sn}} \right) = {- \frac{{\underset{\_}{V}}_{n2}}{{\underset{\_}{I}}_{n2}}}}\mspace{79mu} {{where}\text{:}}}} & (50) \\{{\underset{\_}{V}}_{n2} = {{\underset{\_}{V}}_{{T{({n - 1})}}2}^{{transf}.} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot {\underset{\_}{Z}}_{2{L{({{2n} - 3})}}} \cdot {\underset{\_}{I}}_{{T{({n - 1})}}{n2}}^{{transf}.}} - {d_{({{2n} - 3})} \cdot {\underset{\_}{Z}}_{2{L{({{2n} - 3})}}} \cdot \left( {{\underset{\_}{I}}_{{T{({n - 1})}}{n2}}^{{transf}.} - {\underset{\_}{I}}_{F2}} \right)}}} & (51)\end{matrix}$

10′. Impedance of k^(th) equivalent source (Z_(2Sk)) is calculatedassuming that the fault is located in tapped line section L(2k−2),(actions performed in operations 660 c):

$\begin{matrix}{\mspace{79mu} {{\left( {\underset{\_}{Z}}_{2{Sk}} \right) = {- \frac{{\underset{\_}{V}}_{k2}}{{\underset{\_}{I}}_{k2}}}}\mspace{79mu} {{where}\text{:}}}} & (52) \\{{\underset{\_}{V}}_{k2} = {{\underset{\_}{V}}_{Tk2}^{{transf}.} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot {\underset{\_}{Z}}_{2{L{({{2k} - 2})}}} \cdot {\underset{\_}{I}}_{Tkk2}^{{transf}.}} - {d_{({{2k} - 2})} \cdot {\underset{\_}{Z}}_{2{L{({{2k} - 2})}}} \cdot \left( {{\underset{\_}{I}}_{Tkk2}^{{transf}.} - {\underset{\_}{I}}_{F2}} \right)}}} & (53)\end{matrix}$

11′. Impedance of equivalent sources (Z_(2Sk)) and (Z_(2S(k+1))) iscalculated on the assumption that the fault is located in the linesection between two tap points Tk−T(k+1), (actions performed inoperations 660 d):

$\begin{matrix}{\mspace{79mu} {{\left( {\underset{\_}{Z}}_{2{Sk}} \right) = {- \frac{{\underset{\_}{V}}_{k2}}{{\underset{\_}{I}}_{k2}}}}\mspace{79mu} {{where}\text{:}}}} & (54) \\{{\underset{\_}{V}}_{k2} = {{\underset{\_}{V}}_{Tk2}^{{transf}.} - {d_{({{2k} - 1})} \cdot {\underset{\_}{Z}}_{2{L{({{2k} - 1})}}} \cdot {\underset{\_}{I}}_{{{TkT}{({k + 1})}}2}^{{transf}.}} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot {\underset{\_}{Z}}_{2{L{({{2k} - 1})}}} \cdot \left( {{\underset{\_}{I}}_{{{TkT}{({k + 1})}}2}^{{transf}.} - {\underset{\_}{I}}_{F2}} \right)} + {{\underset{\_}{Z}}_{2{Lk}}{\underset{\_}{I}}_{k2}}}} & (55) \\{\mspace{79mu} {{\left( {\underset{\_}{Z}}_{2{S{({k + 1})}}} \right) = {- \frac{{\underset{\_}{V}}_{{({k + 1})}2}}{{\underset{\_}{I}}_{{({k + 1})}2}}}}\mspace{79mu} {{where}\text{:}}}} & (56) \\{{\underset{\_}{V}}_{{({k + 1})}2} = {{\underset{\_}{V}}_{Tk2}^{{transf}.} - {d_{({{2k} - 1})} \cdot {\underset{\_}{Z}}_{2{L{({{2k} - 1})}}} \cdot {\underset{\_}{I}}_{{{TkT}{({k + 1})}}2}^{{transf}.}} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot {\underset{\_}{Z}}_{2{L{({{2k} - 1})}}} \cdot \left( {{\underset{\_}{I}}_{{{TkT}{({k + 1})}}2}^{{transf}.} - {\underset{\_}{I}}_{F2}} \right)} + {{\underset{\_}{Z}}_{2{{Lk}{({2k})}}}{\underset{\_}{I}}_{{({k + 1})}2}}}} & (57)\end{matrix}$

12′. It is checked (actions performed in operations 670 a, 670 b, 670 c,670 d) whether the calculated equivalent source impedances (Z_(2S1)),(Z_(2Sn)), (Z_(2SC))_(SUB) _(—) _(C), (Z_(2Sk)), (Z_(2S(k+1))) arecontained in the interval in the first quadrant of the complex plane Z,that is, whether both the real and the imaginary part of the calculatedimpedance is greater than zero. The impedance of equivalent sourcesystems is determined for the negative component for occurrence of thefaults: phase-to-earth, phase-to-phase, double phase-to-earth oralternatively for the incremental positive sequence component. Forthree-phase faults the impedance of equivalent source systems iscalculated for the incremental positive sequence component.

Results which are not contained within the first quadrant suggest thatthey were calculated on a false pre-assumption concerning the locationof the fault on the given line section and the result of the calculationof the distance to the fault point made on that assumption is rejected.The remaining results undergo further treatment except the case whereonly one impedance calculated on the assumption that the fault occurredin the given line section is contained in the first quadrant. The resultof the calculation of the distance to the fault point for that impedanceindicates a correct pre-assumption. This result is considered final(actions performed in operation 680).

13′. If the criteria analysis performed as in action 12′ does notprovide an explicit conclusion about which values define the location ofthe fault, then, in subsequent actions (actions performed in operations690 a, 690 b, 690 c, 690 d), the calculated equivalent source impedancesare transformed into a modular form, whereupon the correct result isselected on the basis of the equivalent source impedance module.

14′. Operation 700. If the calculated value of the equivalent sourcesystem impedance module, assuming the occurrence of the fault on thegiven line section, does not correspond to the real value of the sourcesystem impedance module, this means that the preliminary data concerningthe fault location in the given section have been assumed falsely andthe result of the calculation of the distance to the fault point basedon this assumption is rejected. If the calculated value of theequivalent source system impedance module, assuming the occurrence ofthe fault on the given line section, corresponds to the real value ofthe module of impedance of the source system, then the result of thecalculation of the distance to the fault confirms the correctpre-assumption and this result is considered final.

The network of actions shown in FIG. 19 includes the following actionsfor the implementation of the invention:

-   -   measurement of currents and voltages according to point 1′ of        the invention embodiment example,    -   determination of the symmetrical components of measured currents        and voltages and calculation of the total fault current        according to point 2′ of the invention embodiment example,    -   calculation of the consecutive hypothetical distances to the        fault points and the fault resistance assuming that the fault is        located in the first line section L1, the terminal line section        L(2n−3), sections of tapped lines (2k−2) and sections between        consecutive tap point (2k−1), according to points:        3.1.a′-3.2.a′, 3.1.b′-3.4.b′, 3.1.c′-3.3.c′, 3.1.d′-3.3.d′ of        the invention embodiment example    -   checking whether the particular hypothetical distances are        contained in the interval from 0 to 1 in relative units and        rejection of those hypothetical distances whose values are        negative or bigger than 1, according to point 4 of the example        embodiment of the invention.    -   checking whether the values of the fault resistance are bigger        than or equal to zero and rejection of values less than zero,        according to point 5′ of the invention embodiment example,    -   calculation of the impedance of equivalent sources of individual        sections assuming that the fault is located in the given        section, as per points 8′-11′ of the invention embodiment        example,    -   selection of the correct result, according to point 12′ of the        invention embodiment example by rejecting those hypothetical        distances for which impedances of calculated equivalent sources        are not within the first quadrant of the complex system of        co-ordinates,    -   selection of the final result according to point 14′ of the        invention embodiment example, by rejecting those hypothetical        distances for which the calculated value of the module of        impedance of equivalent source systems does not correspond to        the real value of the module of the equivalent source system.

The described example refers to a double phase-to-earth fault of thetype (a-b-g). Yet this method is analogous for other types of faults. Ifother types of faults are analysed, the relevant coefficients a_(F1),a_(F2), a_(F0), a₁, a₂, a₀ change. The values of these coefficients arecompiled in tables 5-6. The method for fault location in multi-terminalelectric power transmission lines according to the present inventioncovers also other types of faults i.e. (a-g, b-g, c-g, a-b, b-c, c-a,b-c-g, c-a-g, a-b-c, a-b-c-g).

The inventive method is not restricted to one line model presented inthe example of the analysis, but it can apply to another model, notshown in the figure, e.g. a long line model. In such case the equations(33-57) will be modified.

The inventive method uses synchronous measurements of currents in allstations of the transmission or distribution system, additionally ituses voltage measurement in the station where the fault locator isinstalled.

The selection of the valid result is based on the aggregation of thethree calculated criterion-type quantities: distance to the fault point,fault resistance in the fault point and impedance of equivalent sourcesystems for those stations where voltage measurement is not done. Thisthird criterion is a two-stage one, i.e. first it is checked whether thedetermined impedances of equivalent source systems are in the firstquadrant of the complex plane Z and then their modules are determined.This first element of the third criterion is innovative and has not beenknown till now. Its advantage is that knowledge of the equivalent sourcesystem impedance is not required for its implementation. Therefore,inaccurate knowledge of these impedances, which is normal in practice,is of no consequence.

1. A method for fault location in electric power lines, in whichdivision of the transmission or distribution system line into sectionsis used and a hypothetical location of the fault is assumed in at leastone of these sections, characterised in that: current for faultcondition and pre-fault condition is measured in all terminal stationsof the system, the line phase voltage for fault and pre-fault conditionsis measured in one terminal station of the system, symmetricalcomponents of the measured current and voltage signals and the totalfault current in the fault point are calculated, the first hypotheticalfault point located in line section between the beginning of the lineand the first tap point, the second hypothetical fault point located inline section between the end of the line and the last tap point of thebranch, and a consecutive hypothetical fault point which is located inthe branch are assumed, while for multi-terminal lines consecutivehypothetical fault points located in the line sections between twoconsecutive tap points are additionally assumed, the distance from thebeginning of the line to the fault point, the distance from the end ofthe line to the fault point, the distance from the end of the tappedline to the fault point located in this branch are calculated, while formulti-terminal lines the distance from the tap point to the fault pointlocated in the line section between two tap points is additionallycalculated, and then for all hypothetical fault points in each sectionfault resistance is calculated, the actual fault point location isselected first by comparing the numerical values concerning thepreviously determined distances and rejecting the results whosenumerical values are negative or bigger than 1 in relative units andthen by analysing the values of the calculated fault resistances forfault points and rejecting those results of the calculations for whichthe value of fault resistance is negative, and if it is found that onlyone numerical value concerning the distance is contained in thenumerical interval between zero and one in relative units and the valueof calculated fault resistance for this distance to the fault point ispositive or equal to zero, then these results are final and theyindicate the actual distance to the fault point and the value of faultresistance in the fault point, if, after the selection of the actualfault point it turns out that at least two numerical values concerningthe previously calculated distances are contained within the numericalinterval from zero to one in relative units and the values of thecalculated fault resistances for these fault points are positive orequal to zero, then impedance modules or impedances of equivalent sourcesystems for the negative sequence component for phase-to-ground faults,phase-to-phase faults, and double phase-to-ground faults or for theincremental positive sequence component for three-phase faults aredetermined and assuming that the fault occurred in a definite section,and during the impedance determination it is additionally checkedwhether the calculated values of the impedance of equivalent sourcesystems are contained in the first quadrant of the Cartesian co-ordinatesystem for the complex plain and these distances to fault points arerejected for which impedance values are not contained in this quadrantof the system, and if it turns out that only one value of the impedanceof the equivalent source system concerning distance is contained in thefirst quadrant of the system, then the result of the calculation of thedistance to the fault point, for this impedance, is considered to befinal, whereas if it turns out that at least two values of the impedanceof equivalent source systems concerning distance are contained in thefirst quadrant of the system, then the modules of these impedances aredetermined, the values of the modules of impedance of equivalent sourcesystems are compared with realistic values which really define thesystem load or supply, and the distance for which the value of themodule of the equivalent source impedance is nearest to the realisticvalues really determining the system load or supply is considered to bethe final result.
 2. A method according to claim 1, characterised inthat the calculation of the total fault current is done taking intoaccount the share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated, a specially determined set ofthese coefficients being used for that operation.
 3. A method accordingto claim 1, characterised in that for double phase-to-earth faults thepositive component is eliminated in the estimation of the total faultcurrent, and for the negative and zero components the following valuesof the share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated are assumed:${{\underset{\_}{a}}_{F\; 1} = 0},{{\underset{\_}{a}}_{F\; 2} = {{\underset{\_}{a}}_{F\; 2}^{{init}.} - \frac{{\underset{\_}{a}}_{F\; 1}^{{init}.}{\underset{\_}{b}}_{F\; 2}}{{\underset{\_}{b}}_{F\; 1}}}},{{\underset{\_}{a}}_{F\; 0} = \frac{{\underset{\_}{a}}_{F\; 1}^{{init}.}}{{\underset{\_}{b}}_{F\; 1}}}$where: a_(F1) ^(init.), a_(F2) ^(init.), a_(F0) ^(init.)—denote theinitial share coefficients determining the relation between thesymmetrical components of the total fault current when voltage dropacross the fault resistance is estimated, b_(F1), b_(F2)—denote relationshare coefficients, determined from the relation between the zerocomponent and the other components of the total fault current flowingthrough the fault resistance.
 4. A method according to claim 1,characterised in that for three-terminal power lines, the distances formthe beginning of the line to fault point d_(A), from the end of the lineto fault point d_(B), from the end of the tapped line to fault pointd_(C) are determined from the following equations:${d_{A} = \frac{{{{real}\left( {\underset{\_}{V}}_{Ap} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {\underset{\_}{V}}_{Ap} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}{{{{real}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},{d_{B} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{imag}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}}} \right){{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}},{d_{C} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{Tp} - {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}},$ where: “real” denotes the real part of the givenquantity, “imag” denotes the imaginary part of the given quantity,V_(Ap)—denotes the fault loop voltage determined assuming that the faultoccurred in section LA, V_(Tp)—denotes the fault loop voltage determinedassuming that the fault occurred in section LB or LC, I_(Ap)—denotes thefault loop current determined assuming that the fault occurred insection LA, I_(TBp)—denotes the fault loop current determined assumingthat the fault occurred in section LB, I_(TCp)—denotes the fault loopcurrent determined assuming that the fault occurred in line section LC,I_(F)—denotes total fault current, Z_(1LA)=R_(1LA)+jω₁L_(1LA)—denotesimpedance of the line section LA for the positive sequence,Z_(1LB)=R_(1LB)+jω₁L_(1LB)—denotes impedance of the line section LB forthe positive sequence, Z_(1LC)=R_(1LC)+jω₁L_(1LC)—denotes impedance ofthe line section LC for the positive sequence, R_(1LA), R_(1LB),R_(1LC)—resistance for the positive sequence for line sections LA, LB,LC, respectively, L_(1LA), L_(1LB), L_(1LC)—inductance for the positivesequence for line sections LA, LB, LC, respectively, ω₁—pulsation forthe fundamental frequency.
 5. A method according to claim 1,characterised in that for three-terminal power lines, the faultresistance R_(FA), R_(FB), R_(FC) is determined from the followingequations: ${R_{FA} = {\frac{1}{2}\begin{bmatrix}{\frac{{{real}\left( {\underset{\_}{V}}_{Ap} \right)} - {d_{A}{{real}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}}}{{real}\left( {\underset{\_}{I}}_{F} \right)} +} \\\frac{{{imag}\left( {\underset{\_}{V}}_{Ap} \right)} - {d_{A}{{imag}\left( {{\underset{\_}{Z}}_{1{LA}}{\underset{\_}{I}}_{Ap}} \right)}}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}\end{bmatrix}}},{R_{FB} = \begin{matrix}{{\frac{1}{2}\left\lbrack \frac{{{real}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{B}} \right) \cdot {{real}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} +} \\{\frac{1}{2}\left\lbrack \frac{{{imag}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{B}} \right) \cdot {{imag}\left( {{\underset{\_}{Z}}_{1{LB}}{\underset{\_}{I}}_{TBp}} \right)}}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}\end{matrix}}$ $R_{FC} = \begin{matrix}{{\frac{1}{2}\left\lbrack \frac{{{real}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{C}} \right) \cdot {{real}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} +} \\{\frac{1}{2}\left\lbrack \frac{{{imag}\left( {\underset{\_}{V}}_{Tp} \right)} - {\left( {1 - d_{C}} \right) \cdot {{imag}\left( {{\underset{\_}{Z}}_{1{LC}}{\underset{\_}{I}}_{TCp}} \right)}}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}\end{matrix}$ where: “real” denotes the real part of the given quantity,“imag” denotes the imaginary part of the given quantity, V_(Ap)—denotesthe fault loop voltage calculated assuming that the fault occurred insection LA, V_(Tp)—denotes the fault loop voltage calculated assumingthat the fault occurred in section LB or LC, I_(Ap)—denotes the faultloop current calculated assuming that the fault occurred in section LA,I_(TBp)—denotes the fault loop current calculated assuming that thefault occurred in section LB, I_(TCp)—denotes the fault loop currentcalculated assuming that the fault occurred in line section LC,I_(F)—denotes the total fault current,Z_(1LA)=R_(1LA)+jω₁L_(1LA)—denotes impedance of the line section LA forthe positive sequence, Z_(1LB)=R_(1LB)+jω₁L_(1LB)—denotes impedance ofthe line section LB for the positive sequence,Z_(1LC)=R_(1LC)+jω₁L_(1LC)—denotes impedance of the line section LC forthe positive sequence, R_(1LA), R_(1LB), R_(1LC)—resistance for thepositive sequence for line sections LA, LB, LC, respectively, L_(1LA),L_(1LB), L_(1LC)—inductance for the positive sequence for line sectionsLA, LB, LC, respectively, ω₁—pulsation for the fundamental frequency.d_(A)—denotes the distance from the beginning of the line to the faultpoint, d_(B)—denotes the distance from the end of the line to the faultpoint, d_(C)—denotes the distance from the end of the tapped line to thefault point.
 6. A method according to claim 1, characterised in that forthree-terminal power lines, the equivalent source impedance for thenegative sequence component ((Z_(2SB))_(SUB) _(—) _(A)) and for theincremental positive sequence component ((Z_(Δ1SB))_(SUB) _(—) _(A)) arecalculated assuming that the fault occurred in LA line section, as perthis equation:${\left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ A} = \frac{{{\underset{\_}{G}}_{iA}{\underset{\_}{I}}_{A\; 2}} - {{\underset{\_}{H}}_{iA}{\underset{\_}{I}}_{FAi}}}{{\underset{\_}{Q}}_{BCi}\left( {{\underset{\_}{I}}_{FAi} - {\underset{\_}{I}}_{Ai}} \right)}},$where: the lower index i takes on values i=2 for the negative sequence,i=Δ1 for the incremental positive sequence component, G_(iA)—denotes thefirst analytical coefficient for the negative sequence component,determined from the analysis of an equivalent circuit diagram of thesystem as shown in FIG. 11 and/or for the incremental positive sequencecomponent analytically determined from the equivalent circuit diagram ofthe system as shown in FIG. 12, I_(Ai)—denotes the negative and/orincremental positive sequence component of current measured at thebeginning of the line, H_(iA)—denotes the second analytical coefficientfor the negative sequence component, determined from the analysis of anequivalent circuit diagram of the system as shown in FIG. 11 and/or theincremental positive sequence component analytically determined from theequivalent circuit diagram of the system as shown in FIG. 12,I_(FAi)—denotes the negative sequence component of the total faultcurrent, determined from the analysis of an equivalent circuit diagramof the system as shown in FIG. 11 and/or the incremental positivesequence component of the total fault current, determined from theanalysis of an equivalent circuit diagram of the system as shown in FIG.12, Q_(BCi)—denotes the quotient of the negative sequence component ofcurrent measured at the end of the line and the sum of the negativesequence components of current signals measured at the end of the lineand at the end of the tapped line and/or the quotient of the incrementalpositive sequence component of current measured at the end of the lineand the sum of the incremental positive sequence components of currentsignals measured at the end of the line and at the end of the tappedline.
 7. A method according to claim 1, characterised in that forthree-terminal power lines, the equivalent source impedance((Z_(2SC))_(SUB) _(—) _(A)) for the negative component and((Z_(Δ1SC))_(SUB) _(—) _(A)) for the incremental positive sequencecomponent are determined assuming that the fault occurred in linesection LA, from the following equation:$\left( {\underset{\_}{Z}}_{iSC} \right)_{SUB\_ A} = {{\left( {{\underset{\_}{Z}}_{iLB} + \left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ A}} \right)\frac{{\underset{\_}{I}}_{Bi}}{{\underset{\_}{I}}_{Ci}}} - {\underset{\_}{Z}}_{iLC}}$where: the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,(Z_(iSB))_(SUB) _(—) _(A)—denotes equivalent source impedance for thenegative sequence component and/or the incremental positive sequencecomponent, calculated assuming that the fault occurred in line sectionLA, Z_(iLB)—denotes the impedance of line section LB for the negativesequence component and/or the positive sequence component, where:Z_(Δ1LB)=Z_(1LB), Z_(1LB)—denotes the impedance of line section LB forthe positive sequence component, Z_(iLC)—denotes the impedance of linesection LC for the negative sequence component and/or impedance of linesection LC for the incremental positive sequence component, whereZ_(2LC)=Z_(1LC) and Z_(Δ1LC)=Z_(1LC), Z_(1LC)—denotes the impedance ofline section LC for the positive sequence component, I_(Bi)—denotes thenegative sequence component and/or the incremental positive sequencecomponent of current measured at the end of the line, I_(Ci)—denotes thenegative sequence component and/or the incremental positive sequencecomponent of current measured at the end of the branch.
 8. A methodaccording to claim 1, characterised in that for three-terminal powerlines, equivalent source impedance for the negative sequence component(Z_(2SB))_(SUB) _(—) _(B) and for the incremental positive sequencecomponent (Z_(Δ1SB))_(SUB) _(—) _(B) is determined assuming that thefault occurred in line section LB, from the following equation:${\left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ B} = \frac{{\left( {1 - d_{B}} \right){\underset{\_}{Z}}_{iLB}{\underset{\_}{I}}_{TBi}^{{transf}.}} - {d_{B}{\underset{\_}{Z}}_{iLB}{\underset{\_}{I}}_{Bi}} - {\underset{\_}{V}}_{Ti}^{{tranf}.}}{{\underset{\_}{I}}_{Bi}}},$where: the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,d_(B)—denotes the distance from the end of the line to the fault point,Z_(iLB)—denotes the impedance of line section LB for the negativesequence component and/or for the positive sequence component, whereZ_(2LB)=Z_(1LB) and Z_(Δ1LB)=Z_(1LB), Z_(1LB)—denotes the impedance ofline section LB for the positive sequence component, I_(TBi)^(transf.)—denotes current flowing from a tap point T to line section LBfor the negative sequence component and/or for the incremental positivesequence component, I_(Bi)—denotes the negative sequence componentand/or the incremental positive sequence component of current measuredat the end of the line, V_(Ti) ^(transf.)—denotes voltage in the tappoint T for the negative sequence component and/or for the incrementalpositive sequence component.
 9. A method according to claim 1,characterised in that for three-terminal power lines, equivalent sourceimpedance for the negative sequence component (Z_(2SC))_(SUB) _(—) _(B)and for the incremental positive sequence component (Z_(Δ1SC))_(SUB)_(—) _(B) is calculated assuming that the fault occurred in line sectionLB, from the following equation:$\left( {\underset{\_}{Z}}_{iSC} \right)_{SUB\_ B} = {- \frac{{\underset{\_}{V}}_{Ci}}{{\underset{\_}{I}}_{Ci}}}$where: the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,V_(Ci)—denotes the negative sequence component and/or the incrementalpositive sequence component of voltage at the end of the tapped line,I_(Ci)—denotes the negative sequence component and/or the incrementalpositive sequence component of current measured at the end of thebranch.
 10. A method according to claim 1, characterised in that forthree-terminal power lines, equivalent source impedance for the negativesequence component (Z_(2SC))_(SUB) _(—) _(C) and for the incrementalpositive sequence component (Z_(Δ1SC))_(SUB) _(—) _(C) is calculatedassuming that the fault occurred in line section LC, from the followingequation:${\left( {\underset{\_}{Z}}_{iSC} \right)_{SUB\_ C} = \frac{{\left( {1 - d_{C}} \right){\underset{\_}{Z}}_{iLC}{\underset{\_}{I}}_{TCi}^{{tranf}.}} - {d_{C}{\underset{\_}{Z}}_{iLC}{\underset{\_}{I}}_{Ci}} - {\underset{\_}{V}}_{Ti}^{{transf}.}}{{\underset{\_}{I}}_{Ci}}},$where: the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,d_(C)—denotes the distance from the end of the tapped line to the faultpoint, Z_(iLC)—denotes impedance of the line section LC for the negativesequence component and/or for the incremental positive sequencecomponent, where Z_(2LC)=Z_(1LC) and Z_(Δ1LC)=Z_(1LC), Z_(1LC)—denotesimpedance of the line section LC for the positive sequence component,I_(TCi) ^(transf.)—denotes current flowing to the tap point T from linesection LC for the negative sequence component and/or for theincremental positive sequence component, I_(Ci)—denotes the negativesequence component and/or the incremental positive sequence component ofcurrent measured at the end of the branch, V_(Ti) ^(transf.)—denotesvoltage at the tap point T for the negative sequence component and/orfor the incremental positive sequence component.
 11. A method accordingto claim 1, characterised in that for three-terminal power lines,equivalent source impedance for the negative sequence component(Z_(2SB))_(SUB) _(—) _(C) and for the incremental positive sequencecomponent (Z_(Δ1SB))_(SUB) _(—) _(C) is calculated assuming that thefault occurred in line section LC, from the following equation:$\left( {\underset{\_}{Z}}_{iSB} \right)_{SUB\_ C} = {- \frac{{\underset{\_}{V}}_{Bi}}{{\underset{\_}{I}}_{Bi}}}$where: the lower index i takes on values i=2 for the negative sequencecomponent, i=Δ1 for the incremental positive sequence component,V_(Bi)—denotes the negative sequence and/or the incremental positivesequence component of voltage at the end of the line, I_(Bi)—denotes thenegative sequence and/or the incremental positive sequence component ofcurrent measured at the end of the line.
 12. A method according to claim1, characterised in that for multi-terminal power lines, distances fromthe beginning of the line to the fault point (d₁), from the end of theline to the fault point (d_((2n−3))), from the end of the line to thefault point (d_((2k−2))), from the tap point to the fault point in theline section between two tap points (d_((2k−1))) are determined from thefollowing equations:${d_{1} = \frac{{{{real}\left( {\underset{\_}{V}}_{1p} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {\underset{\_}{V}}_{1p} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}{{{{real}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} - {{{imag}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}}},{d_{({{2n} - 3})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{{T{({n - 1})}}{np}} - {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}}$ $d_{({{2k} - 2})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{Tkkp} - {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{Tkkp} - {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}$ ${d_{({{2k} - 1})} = \frac{\begin{matrix}{{{- {{real}\left( {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}}} \right)}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} +} \\{{{imag}\left( {{\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} - {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}{\begin{matrix}{{{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}{{imag}\left( {\underset{\_}{I}}_{F} \right)}} -} \\{{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}{{real}\left( {\underset{\_}{I}}_{F} \right)}}\end{matrix}}},$ where: V_(1p)—fault loop voltage calculated underassumption that fault occurred in the first section of the line sectionL1, I_(1p)—fault loop current calculated under assumption that faultoccurred in the first section of the line section L1, V_(T(n−1)np)—faultloop voltage calculated under assumption that fault occurred in the linesection L(2n−3), I_(T(n−1)np)—fault loop current calculated underassumption that fault occurred in the line section L(2n−3),V_(Tkkp)—fault loop voltage calculated under assumption that faultoccurred in the k^(th) tapped line, I_(Tkkp)—fault loop currentcalculated under assumption that fault occurred in k^(th) tapped line,V_(TkT(k+1)p)—fault loop voltage calculated under assumption that faultoccurred in the line section between two tap points, I_(TkT(k+1)p)—faultloop current calculated under assumption that fault occurred in the linesection between two tap points, I_(F)—total fault current,Z_(1L1)—impedance of line section L1 for the positive sequencecomponent, Z_(0L1)—impedance of line section L1 for the zero sequencecomponent, Z_(1L(2n−3))—impedance of line section L(2n−3) for thepositive sequence component, Z_(0L(2n−3))—impedance of line sectionL(2n−3) for the zero sequence component, Z_(1L(2k−2))—impedance of linesection L(2k−2) for the positive sequence component,Z_(0L(2k−2))—impedance of line section L(2k−2) for the zero sequencecomponent, Z_(1L(2k−1))—impedance of line section L(2k−1) for thepositive sequence component, Z_(0L(2k−1))—impedance of line sectionL(2k−1) for the zero sequence component. k—number of the tap pointn—number of the line terminal
 13. A method according to claim 1,characterised in that for multi-terminal electric power lines, faultresistance (R_(1F)), (R_((2n−3)F)), (R_((2k−2)F)), (R_((2k−1)F)) isdetermined from the following equations:${R_{1F} = {\frac{1}{2}\begin{bmatrix}{\frac{{{real}\left( {\underset{\_}{V}}_{1p} \right)} - {d_{1}{{real}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}}}{{real}\left( {\underset{\_}{I}}_{F} \right)} +} \\\frac{{{imag}\left( {\underset{\_}{V}}_{1p} \right)} - {d_{1}{{imag}\left( {{\underset{\_}{Z}}_{1L\; 1}{\underset{\_}{I}}_{1p}} \right)}}}{{imag}\left( {\underset{\_}{I}}_{F} \right)}\end{bmatrix}}},{R_{{({{2n} - 3})}F} = \begin{matrix}{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{{T{({n - 1})}}{np}} \right)} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2n} - 3})}}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} +} \\{{+ {\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{{T{({n - 1})}}{np}} \right)} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{({{2n} - 3})}}{\underset{\_}{I}}_{{T{({n - 1})}}{np}}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}},}\end{matrix}}$ $R_{{({{2k} - 2})}F} = \begin{matrix}{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{Tkkp} \right)} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} +} \\{+ {\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{Tkkp} \right)} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 2})}}}{\underset{\_}{I}}_{Tkkp}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}\end{matrix}$ $R_{{({{2k} - 1})}F} = \begin{matrix}{{\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{real}\left( {\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} \right)} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot}} \\{{real}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}\end{matrix}}{{real}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack} +} \\{+ {\frac{1}{2}\left\lbrack \frac{\begin{matrix}{{{imag}\left( {\underset{\_}{V}}_{{{TkT}{({k + 1})}}p} \right)} - {\left( {1 - d_{({{2k} - 1})}} \right) \cdot}} \\{{imag}\left( {{\underset{\_}{Z}}_{1{L{({{2k} - 1})}}}{\underset{\_}{I}}_{{{TkT}{({k + 1})}}p}} \right)}\end{matrix}}{{imag}\left( {\underset{\_}{I}}_{F} \right)} \right\rbrack}}\end{matrix}$ where: d₁—distance to fault from the beginning of the lineto the fault point, d_((2n−3))—distance to fault from the end of theline to the fault point, d_((2k−2))—distance to fault from the end ofthe tapped line to the fault point d_((2k−1))—distance to fault in theline section between two tap points, V_(1p)—fault loop voltagecalculated under assumption that fault occurred in the first section ofthe line section L1, I_(1p)—fault loop current calculated underassumption that fault occurred in the first section of the line sectionL1, V_(T(n−1)np)—fault loop voltage calculated under assumption thatfault occurred in the line section L(2n−3), I_(T(n−1)np)—fault loopcurrent calculated under assumption that fault occurred in the linesection L(2n−3), V_(Tkkp)—fault loop voltage calculated under assumptionthat fault occurred in the k^(th) tapped line, I_(Tkkp)—fault loopcurrent calculated under assumption that fault occurred in k^(th) tappedline, V_(TkT(k+1)p)—fault loop voltage calculated under assumption thatfault occurred in the line section between two tap points,I_(TkT(k+1)p)—fault loop current calculated under assumption that faultoccurred in the line section between two tap points, I_(F)—total faultcurrent, Z_(1L1)—impedance of line section L1 for the positive sequencecomponent, Z_(0L1)—impedance of line section L1 for the zero sequencecomponent, Z_(1L(2n−3))—impedance of line section L(2n−3) for thepositive sequence component, Z_(0L(2n−3))—impedance of line sectionL(2n−3) for the zero sequence component, Z_(1L(2k−2))—impedance of linesection L(2k−2) for the positive sequence component,Z_(0L(2k−2))—impedance of line section L(2k−2) for the zero sequencecomponent, Z_(1L(2k−1))—impedance of line section L(2k−1) for thepositive sequence component, Z_(0L(2k−1))—impedance of line sectionL(2k−1) for the zero sequence component. k—number of the tap pointn—number of the line terminal
 14. A method according to claim 1,characterised in that for multi-terminal power lines, equivalent sourceimpedance for the negative sequence component (Z_(2S1)) or for theincremental positive sequence component (Z_(Δ1S1)) is calculatedassuming that the fault is located in line section between the beginningof the line and the first tap point, from the following equation:$\left( {\underset{\_}{Z}}_{{iS}\; 1} \right) = \frac{- {\underset{\_}{V}}_{1i}}{{\underset{\_}{I}}_{1i}}$where i=2 for negative sequence, Δ1 for incremental positive sequencecomponent, V_(1i)—voltage measured in station 1 (the first lower index)for individual symmetrical components, (the second lower index) i.e.negative component—index 2 and incremental positive sequencecomponent—index Δ1, I_(1i)—current measured in station 1 (the firstlower index) for individual symmetrical components, (the second lowerindex) i.e. negative component—index 2 and incremental positive sequencecomponent—index Δ1.
 15. A method according to claim 1, characterised inthat for multi-terminal power lines equivalent source impedance((Z_(2S(n)))) for the negative component and ((Z_(Δ1S(n)))) for theincremental positive sequence component is determined assuming that thefault is located in line section between the end of the line and thefinal tap point, from the following equation:$\left( {\underset{\_}{Z}}_{iSn} \right) = {- \frac{\begin{matrix}{{\underset{\_}{V}}_{{T{({n - 1})}}i}^{{transf}.} - {\left( {1 - d_{({{2n} - 3})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2n} - 3})}} \cdot}} \\{{\underset{\_}{I}}_{{T{({n - 1})}}{ni}}^{{trans}.} - {d_{({{2n} - 3})} \cdot {\underset{\_}{Z}}_{{iL}{({{2n} - 3})}} \cdot \left( {{\underset{\_}{I}}_{{T{({n - 1})}}{ni}}^{{tranf}.} - {\underset{\_}{I}}_{Fi}} \right)}}\end{matrix}}{{\underset{\_}{I}}_{ni}}}$ where: i=2 for negativesequence component, Δ1 for incremental positive sequence component,V_(T(n−1)i) ^(transf.)—voltages in the final tap point T(n−1) for thenegative sequence component i=2, or for incremental positive sequencecomponent i=Δ1, d_((2n−3))—distance to fault from the end of the line tothe fault point, Z_(iL(2n−3))—impedance of line section L(2n−3) for thenegative sequence component i=2, or for incremental positive sequencecomponent i=Δ1, I_(T(n−1)ni) ^(transf.)—values of current flowing fromtap point T(n−1) to station n in line section L(2n−3) for negativesequence component i=2, or for incremental positive sequence componenti=Δ1 I_(Fi)—total fault current for negative sequence component i=2, orfor incremental positive sequence component i=Δ1, I_(ni)—currentmeasured in last station n (the first lower index) for individualsymmetrical components, (the second lower index) i.e. negativecomponent—index 2 and incremental positive sequence component—index Δ1.16. A method according to claim 1, characterised in that formulti-terminal power lines equivalent source impedance for the negativecomponent ((Z_(2Sk))) and for the incremental positive sequencecomponent ((Z_(Δ1Sk))) is determined assuming that the fault is locatedin the line branch:$\left( {\underset{\_}{Z}}_{iSk} \right) = {- {\frac{\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{tranf}.} - {\left( {1 - d_{({{2k} - 2})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 2})}} \cdot {\underset{\_}{I}}_{Tkki}^{{tranfs}.}} -} \\{d_{({{2k} - 2})} \cdot {\underset{\_}{Z}}_{{iL}{({{2K} - 2})}} \cdot \left( {{\underset{\_}{I}}_{Tkki}^{{transf}.} - {\underset{\_}{I}}_{F\; 2}} \right)}\end{matrix}}{{\underset{\_}{I}}_{ki}}.}}$ where: V_(Tki)^(transf.)—voltages in the k^(th) tap point for the negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1,d_((2k−2))—distance to fault from the end of the tapped line to thefault point Tk, Z_(iL(2k−2))—impedance of line section L(2k−2) for thenegative sequence component i=2, or for incremental positive sequencecomponent i=Δ1, I_(Tkki) ^(transf.)—values of current flowing from tappoint Tk to k^(th) station in tapped line section L(2k−2) for negativesequence component i=2, or for incremental positive sequence componenti=Δ1 I_(Fi)—total fault current for negative sequence component i=2, orfor incremental positive sequence component i=Δ1, I_(ki) currentmeasured in station k (the first lower index) for individual symmetricalcomponents, (the second lower index) i.e. negative component—index 2 andincremental positive sequence component—index Δ1.
 17. A method accordingto claim 1, characterised in that for multi-terminal power linesequivalent source impedance for the negative component ((Z_(2Sk)) and(Z_(2S(k+1)))) as well as for the incremental positive sequencecomponent ((Z_(Δ1Sk)) and (Z_(Δ1S(k+1)))) is determined assuming thatthe fault is located in the line section between two consecutive tappoints, from the following equation:$\left( {\underset{\_}{Z}}_{iSk} \right) = {- \frac{\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{transf}.} - {d_{({{2k} - 1})} \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot {\underset{\_}{I}}_{{{TkT}{({k + 1})}}i}^{{transf}.}} -} \\{{\left( {1 - d_{({{2k} - 1})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot \left( {{\underset{\_}{I}}_{{{TkT}{({k + 1})}}i}^{{transf}.} - {\underset{\_}{I}}_{Fi}} \right)} + {{\underset{\_}{Z}}_{iLk}{\underset{\_}{I}}_{ki}}}\end{matrix}}{{\underset{\_}{I}}_{ki}}}$$\left( {\underset{\_}{Z}}_{{iS}{({k + 1})}} \right) = {- \frac{\begin{matrix}{{\underset{\_}{V}}_{Tki}^{{tranf}.} - {d_{({{2k} - 1})} \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot {\underset{\_}{I}}_{{{TkT}{({k + 1})}}i}^{{transf}.}} -} \\{{\left( {1 - d_{({{2k} - 1})}} \right) \cdot {\underset{\_}{Z}}_{{iL}{({{2k} - 1})}} \cdot \left( {{\underset{\_}{I}}_{{{TkT}{({k + 1})}}i}^{{transf}.} - {\underset{\_}{I}}_{Fi}} \right)} + {{\underset{\_}{Z}}_{{iL}{({2k})}}{\underset{\_}{I}}_{{({k + 1})}i}}}\end{matrix}}{{\underset{\_}{I}}_{{({k + 1})}i}}}$ where: V_(Tki)^(transf.)—voltages in the k^(th) tap point for the negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1,d_((2k−1))—distance to fault in the line section between two tap pointsZ_(iL(2k−1))—impedance of line section L(2k−1) for the negative sequencecomponent i=2, or for incremental positive sequence component i=Δ1,I_(TkT(k+1)i) ^(transf.)—current flowing from tap point Tk to tap pointT(k+1) in the line section for negative sequence component i=2, or forincremental positive sequence component i=Δ1 I_(Fi)—total fault currentfor negative sequence component i=2, or for incremental positivesequence component i=Δ1, I_(ki) current measured in station k (the firstlower index) for individual symmetrical components, (the second lowerindex) i.e. negative component—index 2 and incremental positive sequencecomponent—index Δ1, I_((k+1)i)—current measured in station k+1 (thefirst lower index) for individual symmetrical components, (the secondlower index) i.e. negative component—index 2 and incremental positivesequence component—index Δ1